ajCLEAR RELAXATION STUDY OF MOLECULAR
MOTION IN LIQUID AND SOLID AMMONIA
By
JAMES LYNN CAROLAN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1969
ACKNOWLEDGMENTS
Of all the sections of a dissertation, this is the most difficult
to write: Difficult because one must for sake of brevity choose a select
group for recognition from a large number of people who had a profound
influence en the work.
To begin, I wish to express appreciation to the members of my doc
toral committee: Drs. T, A. Scott, A. A. Sroyles, E, H. Hadlock, C. F.
Hooper and J. S. Rosenshain, for their many tangible and intangible
contributions to this work. Special thanks go to the chairman of my
com.mittee, Dr. T. A. Scott, for suggesting this study and providing a
very well equipped laboratory in which to perform the work. I wish to
further thank Dr. Scott for laboring through the rough draft of this
manuscript and offering a multitude of helpful suggestions. Special ap
preciation is also given Dr. S. S. Ballard, the very able chairman of
the Department of Physics, vjho first made this work possible by grant
ing me a departmental assistantship en an unusually late application.
Appreciation is also given my many colleagues vjhose contributions
are manifold. Special thanks go to Mr. P. C. P. Canepa for guidance
J through the initial confrontation with research electronics, for offering
numerous very helpful suggestions throughout the course of this work,
and for use of proton lines that he measured in am:monla at approximately
1 and 4.2Ko I am also indebted to Dr. James Pl. Srookeraan for many
XI
^^•SiAWBMlIf tM>. Ha^ ft^
interesting and informative conversations, and to Mr. Basil McDowell
for very able guidance in the art of cryogenic equipment fabrication.
Among the people who had no influence on what is written in this
work, but a profound influence on the fact that it exists; I wish to
J express appreciation to my parents, Mr. and Mrs. James W. Carolan, and
my wife's parents. Dr. and Mrs. Richard W. Setzer, for continuing en
couragement throughout the course of this work. Very special thanks go
to my wife, Mrs. Joan Setzer Carolan, for encouragement through many
trying periods, and for careful proofreading of the manuscript.
Finally, I wish to thank Mrs. M. Beth Sercombe for her diligent
proofreading and highly competent typing of the final manuscript.
Ill
TABLE OF CONTENTS
^ Page
ACKNOWLEDGMENTS ii
LIST OF TABLES '"^
LIST OF FIGURES "^iii
CHAPTER I INTRODUCTION 1
CHAPTER II THE AMMONIA MOLECULE IN THE SCHEI^E OF NlIR 4
2.1. Molecular Bonding in Ammonia 4
2.2. Motivation for the Use of Nuclear Magnetic
Resonance Techniques in the Study of Molec
ular Motion 6
2.3. The Molecular Hamiltonian 8
2.3.1. The Zeeman Term 8
2.3.2. Nuclear DipoleDipole Interaction
Term 9
2.3.3. The Quadrupole Interaction . 12
2.3.4. Magnetic Interactions of Nuclei with
Molecular Electrons 14
2.4. Summary 17
CHAPTER III INTRODUCTION TO NUCLEAR MAGNETIC RESONANCE AND
RELAXATION 20
3.1. Thermal Equilibrium of the Nuclear Spin System ... 20
3.2. Nuclear Magnetic Resonance Phenomenon 21
3.3. Continuous Wave vs. Transient N14R 26
■r^ 3.3.1. General Remarks 26
^'4^ ^ 3.3.2. Relationship Between the Lineshape and
Free Induction Decay 27
3.3.3. Moments of the Lineshape 34
3.4. The Effect of Molecular Motion on the Free In
duction Decay and Lineshape in Ammonia 35
3.4.1. The FID and Lineshape in Solid IY6. 37
iv
J
Page
3.4.2, The FID and Lineshape in Liquid NH 44
3.5. The SpinLattice Relaxation Time 47
CHAPTER IV INTERPRETATION OF EXPERIMENTAL RESULTS 51
4.1. Outline of Experimental Results 51
4.2. Germane Properties of Liquid and Solid Ammonia ... 52
4.3. Introduction to Nuclear Relaxation via Molec
ular Motion 57
4.4. Interpretation of Relaxation and Continuous
Wave Data from Protons in Solid Ammonia 60
4.4.1. Interpretation of Proton Relaxation
Data
4.4.2. Interpretation of Proton Linewidth Data ... 80
4.4.3. Calculation and Interpretation of the
Proton Second Moment 84
4.5. Interpretation of Nuclear Magnetic Relaxation
Data from 14^ and Protons in Liquid Ammonia 92
4.5.1. Calculation of Molecular Reorientation
Correlation Time from Ti Measurements
on the l^N Nucleus 92
4.5.2. Interpretation of Proton T^ and T2 Data
from Liquid NH , . 100
CHAPTER V INSTRUMENTATION AND MEASUREMENT PROCEDURE 113
5.1. Design of Cryostat and Sample Handling Procedure . . 113
5.1.1. Sample History and Handling Procedure .... 113
5.1.2. Construction and Performance of the
Temperature Control Cryostat 116
5.1.3. Temperature Measurement, Control and
Stability 119
5.2. Measurement Procedures Used . .■ 125
5.2.1. Measurement of T by 90°t90° Pulse
Sequence 125
5.2.2. The Measurement of the SpinSpin Relax
ation Time 129
5.2.3. The Measurement of Linewidth and Second
Moment 135
.5.3. Electronic Apparatus 136
5.3.1. Pulsed Spectrometer 136
Continuous Wave Oscillator 138
Gated Power Amplifier .  138
Pulse Program Circuitry 140
J
Page
Data Recording Device 143
Magnet 143
Frequency and Time Measurements 143
The Coupling Network and Receiver 143
5.3.2. The Continuous Wave Spectrometer "" 150
BIBLIOGRAPHY  • ■■ 153
BIOGRAPHICAL SKETCH 158
J
vx
LIST OF TABLES
Table Page
1. The independent contributions to the proton
second moment for solid NH„ 85
J
VI 1
)
LIST OF FIGURES 
Figure Page
1. Free ammonia molecule 5
2. The laboratory coordinate system 9
3. Laboratory and star coordinate systems 23
4. The rotating components of a linearly polarized
oscillatory magnetic field 23
5. 'Ideal' spin system behavior following a 90° pulse .... 29
6. (a.) Temporal development of M *(t) following a
90° pulse \ 33
(b.) Voltage induced in sample coil 33
(c.) Phase coherent detected sample coil voltage .... 33
7. (a.) FID (protons of NH at 116 K) 39
(b.) Solid NH„ lineshape function at three tem
peratures 39
8.  (a.) Liquid ammonia lineshape at 201 K (from ref.
(19)) 46
(b.) FID expected from liquid ammonia (protons), 46
9. Structure of solid NH 54
EXP '\
10. Proton T vs. 10 /(absolute temperature) 61
EXP "^
11. Proton T vs. 10" / (absolute temperature) 62
EXP 3
12. Proton T vs. 10 /(absolute temperature) 63
13. Proton T^ vs. 10 /(absolute temperature) . 64
14. LN(M recovery) vs. time protons in NH^ 69
15. LN <R(t)> vs. tlme/T' 71
Vlll
Figure Page
16. Theoretical proton T reorientation correlation .... 72
17. Log_ (hindered rotation correlation time x )
vs . temperature 75
18. Proton linewidth vs. temperature 81
19. Second moment vs. temperature 86
20. Log (T^) vs. 10"^/ (temperature) 94
21. Correlation time t vs. temperature 97
22. Proton spinspin relaxation time T„ vs. tem
perature 109
23. Log^Q(T ) vs. 10 /(temperature) Ill
24. Diagram of the gas handling system 114
25. The temperature control cryostat 117
26. Block diagram of the temperature control loop . . .' . . 122
27. Cryostat heater input power vs. sample chamber
temperature 123
28. Sample chamber and temperature bomb 124
29. Temperature difference vs. sample chamber
temperature 126
30. The 90°T90° pulse sequence 128
31. (a.) Sample chamber and magnet coordinate system . . . 132
(b.) Relative FID amplitude (protons NH„) vs.
time 132
32. Block diagram of the pulse spectrometer electronics . . 137
33. (a.) Configuration used for 90°t90° pulse
y sequence 139
(b.) Typical data output from 90°t90'' experi
ment 139
34. (a.) Configuration used in CarrPurcell sequence . . . 142
(b.) Output at points indicated in fig. 34 (a.) .... 142
Ix
'\
J
Figure Page
34. (c.) Typical data output from CarrPurcell sequence . . 142
35. Typical parallel tuned sample coil circuit 148
36. LEL input circuit 148
37. Arenburg input circuit 148
38. Block diagram of the continuous wave apparatus 151
CHAPTER I
INTRODUCTION
The problem to be considered in this dissertation is the use of
experimental nuclear magnetic resonance techniques to arrive at con
clusions concerning the character of molecular motion in ammonia
throughout the temperature range from — 1 to 239 K. More explicitly,
we obtain numbers characterizing defined behavior of the molecular
nuclear spin system under certain welldefined conditions; then attempt
to understand the average microscopic behavior of the molecular system
using theories reported in the scientific literature. This disserta
tion reports work of an experimental nature; thus we avoid reiterating
a large amount of theoretical material which is handled quite well in
the extensive literature on the subject. It has been the author's
experience that the many wellxjritten original publications and re
views of the literature by experts in the field are of much greater
utility than the pseudoreviews one commonly finds in dissertations.
Some theory is of course necessary for selfconsistency, but we will
adhere to a phenomenological approach with heavy reference to articles
j^ and texts that the author finds particularly helpful.
Chap. II is devoted to a discussion of the aimnonia molecule from
the viewpoint of the radiofrequency spectroscopist , i.e., we will
introduce the parts of the molecular Hamiltonian relevant to nuclear
magnetic resonance. We attempt to communicate the motivation one has
j
for using nuclear resonance techniques for a study of this nature.
Nuclear Resonance techniques may be divided into two broad cate
gories: Transient and continuous wave NMR. Within the framework of
continuous wave NMS, the information output of an experiment takes the
form of a lineshape, or frequency distribution, function; whereas in
the case of transient NMR one obtains the same information from the
free induction decay. The intimate inverse relationship in the fre
quency and time domains, respectively, between the lineshape function
and the free induction decay is not often emphasized. We treat this
relationship in some depth in Chap. Ill after introducing the NMR
phenomenon via the standard rotating coordinate treatment. A dis
cussion of the effect of molecular motion on the lineshape function
and free induction decay follows, and is illustrated by a comparison
of theory with experimental results from solid ammonia. We then
attempt to predict, using previously presented theory, how the ideal
free induction decay of protons in liquid ammonia should appear. The
discussion in Chap. Ill is based on the assumption that ideal apparatus
is available, and this should be kept in mind. Instrumental defects
can introduce large errors; for. instance, magnetic field inhomogeneity
is particularly critical in a lineshape study of a liquid, a problem
to be discussed in some detail in Chap. V.
In Chap. IV we present and discuss our experimental results. The
presentation is begun with a general survey of previous experimental
work on ammonia, the results of which we use to corroborate the model
of molecular motion adopted in order to explain our experimental re
sults. Fortunately, rather specific detailed theories, which fit our
models of molecular motion nicely, are available from the scientific
J
literature. Instead of first introducing the theories and then in a
separate section applying a theory to a set of experimental results,
we introduce and give the motivation for use of a particular theory as,
it is required. It is thought that this procedure is more conducive
for communication of the process involved in applying the theoretical
results to our experimental results.
Chap. V is devoted exclusively to experimental procedure and
apparatus. We first discuss the apparatus and procedure directly
connected with handling and temperature control of the sample. The
actual procedure used in taking and processing data is then explained.
Finally, we discuss the electronic apparatus used in the experiment.
For the most part, the basic design and operation of the electronic
apparatus is fairly straightforward, but there exists one very chal
lenging problem which deserves detailed consideration. This is the
problem of mutually coupling the sample coil to both the power ampli
fier and the receiver. Our discussion of this problem is accompanied
by a number of references to the literature in order that the inter
ested reader may further pursue this interesting electronics problem.
CHAPTER II
J THE AMMONIA MOLECULE IN THE SCHEME OF NMR
2.1. Molecular Bonding in Ammonia
The ammonia molecule consists of three hydrogen atoms bonded to a
single nitrogen. Fig. 1 illustrates the molecular geometry for the
isolated molecule (1). Note the C. sjmmietry about the z axis. This
symmetry is preserved in the lattice structure of solid ammonia.
Since the molecular structure is the determining factor for
many of the properties of the compound it seems proper at this point
to consider the construction of the molecule. Pauling (2) in 1931
introduced the idea that the direction of a given covalent bond should
be approximately the direction which yields the maximum overlap of the
wave functions associated with the bonding electrons of the respective
atoms. With this in mind we consider the ground state of the nitrogen
atom. Nitrogen has an atomic number seven and thus three unpaired
electrons in the 2p state. Recall that the p , p , and p orbitals
can be pictured as figure eights of revolution, the axes of which lie
out the X, y, and z axes respectively. Hydrogen has only a single
V electron in the state Is. Thus with Pauling's theory in mind we would
expect the three hydrogens to bond to the nitrogen in such a manner that
each of the 2p electrons forms a spinup, spindown pair with one of
the hydrogen Is electrons. Were this the complete story we would expect
the three NH bonds to be mutually perpendicular. This is not the case,
as is indicated by experiment, and Pauling cites as the cause for the
discrepancy the partial ionic character of the NH bond. This leads
to a net positive charge at each hydrogen and a coulombic repulsion,
which tends to increase the bond angles.
>>
Ca Symmetry Axis
Nitrogen
Hydrogen
Rnh= 1008 A
e = 107
Rhh = '62 A
Fig. I. FREE AMMONIA MOLECULE
J
i
Another important aspect of this effect, one which has far reach
ing consequences, is that the bonded electron wave functions tend to
concentrate closer to the highly electronegative nitrogen. This re
sults in what may be considered a distortion of the 2s orbital forming
what is known as a lone pair (3) of electrons out the +z axis. This
high concentration of electron wave function leads to the formation of
intermolecular bonds of an ionic nature (2) known as hydrogen bonds.
This type of bonding is responsible for the abnormally high boiling
point of liquid ammonia and for the bonding of the solid. The hydrogen
bond energy is estimated by Pauling to be 1.3 kcal/mole. The 93.4
kcal/mole NH bond energy (4) gives one some perspective to the magni
tude of the hydrogen bond energy.
2.2. Motivation for the Use of Nuclear Magnetic Resonance
Techniques in the Study of Molecular Motion
The objective of this experiment is to study the motion of a typi
cal ammonia molecule in the various possible natural physical states
of the pure substance. This is not a trivial task for a multitude of
reasons, but primarily because any measurement we may perform neces
sarily involves an exchange of energy, thereby altering the normal
physical state of the system. We must therefore strike the most favor
able compromise with Nature: We compromise the ability to study a
single molecule and look instead at the average motion of an effective
ly infinite nxjmber in return for using very low energy techniques.
Nature has been kind in this respect in that she provides the proper
ties necessary to use the smallest possible probe, the nucleus. Many
nuclei (5) possess spin angular momentum and a magnetic moment and
those with spin greater than 1/2 also possess an electric quadrupole
14
moment. For example the N nucleus has a maximum observable magnetic
moment of y = .404 in units of the nuclear magneton (y = 5.050 x 10
erg/gauss), a spin 1=1, and an electric quadrupole moment of
—26 2
7.1 X 10 cm . The hydrogen nucleus has a maximum observable mag
) netic moment y = 2.7927 y and a spin I = 1/2. It is found that when
placed in a magnetic field H, the nucleus with magnetic moment y" has
an energy of orientation
E = yH (2.1)
Furthermore, only 21+1 states are observed and have energy eigen
values
E = YhmHi (2.2)
where
m= [1, dl),..., 0, 1] (2.3)
and Y Is a constant of proportionality, called the gyromagnetic ratio,
which is determined empirically. The energy of transition for a proton
in a 5 kG magnetic field from the spindown to the spinup state is
—8
8.8 X 10 eV and the transition energy in the same field for a bare
nitrogen nucleus from say a j 1> to a  0> state is over an order of
magnitude smaller. The electron also possesses a spin I = 1/2 but the
transition energy it requires is about 650 times that of the relatively
strongly coupled proton. Due to the diamagnetic nature of NH_, no
J
strong paramagnetic spin or orbital coupling with the applied field
exists.
We may thus conclude that the application of a magnetic field to a
system of ammonia molecules results in a thoroughly trivial perturba
tion to the molecular motion and the results obtained by using these
techniques are quite characteristic of the unperturbed system.
2.3. The Molecular Hamlltonian
The Hamlltonian of the isolated ammonia molecule immersed in the
magnetic field is to be written in a form suitable for the experiment,
and the individual terms examined in some detail. The general Hamll
tonian for an atomic system is written out explicitly in Slichter (6).
We group all terms in the molecular Hamlltonian which are not relevant
to the problem at hand into the term Ji . Ironically we are most
Si
interested in that part of H describing the motion of the molecule,
but must abandon it experimentally in order that the temporal develop
ment of the part of the wave function in product space governed by
"H is not disturbed by our measurements. In order for this to be
a
true the quantum numbers associated with f\ before we apply the per
3.
turbation to the nuclear spin system must remain good quantum numbers.
As will be seen subsequently the form of the portion of the molecular
Hamlltonian that we work with will provide a back door through which
we are able to extract the desired information.
The remaining, more relevant terms of the molecular Hamlltonian
are now written out. Since ultimately we are interested in describing
molecular motion, we will also briefly consider the effect of transla
tion and rigid body rotation on these terms of the Hamlltonian. We
assume that at the temperatures at which we work the molecules may be
considered as totally rigid.
2.3,1. The Zeeman Term
This term forms the major contribution to the relevant Hamlltonian.
3
z ^, P o 1 'N o N
J=l
The first term in Eq, (2.24) constitutes the total proton contribution
J
to the molecular energy as defined in Eq. (2.1), and the second term
is the contribution due to the nitrogen nucleus. H is the applied
magnetic field. We choose our coordinate system for simplicity such
that H lies out the +z axis. The observable energy eigenvalues are
those given in Eq. (2.2). Rotation of the molecule does not affect this
term, and translation has an effect only if spatial inhomogeneity
exists in the external field. We assume unless otherwise specified
that H is spatially and temporally constant.
2.3.2. Nuclear DipoleDipole Interaction Term
)
4 4
" j=i k=i
dd
^±:!k _ if?£lik^iiki
r\
3
(2.5)
.th
where y . = y.'fil. is the magnetic moment of the i nucleus r., is the
3 J J Jk
vector from nucleus j to nucleus k. This term involves molecular
coordinates and will prove to be the most important term for obtaining
the desired information regarding molecular motion. A laboratory
coordinate system is shotra in Fig. 2 in which the Zeeman Hamiltonian
can be expressed as a particularly simple diagonal matrix if the proper
set of basis functions is chosen. In this same coordinate system (7)
Eq. (2.5) may be expressed in a much more palatable form.
Fig. 2. The Laboratory Coordinate System
10
Eq. (2.5) becomes,
^dd"2 ^ r  ' 3 [A+B+C+D+E+F] (2.6)
3=1 k=l r.^
2
where, A = I. I, (13 cos e., )
jz kz^ jk
^ B =i[i;!"i" + i:iJ;](i3 cos^9.,) = i(i3 cos^e.,)(i. I, i.i)
^Jk jk jk2 jk^jzkzjk
C Ili"!"!, + I. l"^]sin9.,cose. e'^'^jk
Zjkz jzz jk jk
D =[lTl, + I. irjsine., cose. e^*jk
2* J kz jz k' jk jk
E =r I.I, sxn 9 ., e J'^
4 J k jk
F=i:i:sin2e.,e''*Jk
4 jTc jk
and, it = I. +11.
J jx jy
I. = I.  il . are the usual raising and lox^^ering operators.
A particularly illuminating discussion of the individual terms and
their effect on a system of two spin 1/2 particles is given in Slich
ter (6), pp. 4550. Since this term is so important in what follows it
is deemed worthwhile to consider in more detail the effect of each
term of Eq. (2.6) in the case of an isolated pair of identical spin
1 1/2 particles. ^^^ ^sy be considered a perturbation on the Zeeman
Hamiltonian. In support of the use of Eq. (2.6) as a perturbation to
Jx we give some idea of the relative sizes of J~l and J~l, , by calcu
z z dd
J lating the local field one proton in NH„ 'sees' due to a neighboring
proton on the same molecule . •
—8
where r = 1.62 X 10 cm, and
24
y = 2.79 X 5.05 X 10 erg/G.
J
11
Eq. (2.7) gives H = 4,3G whereas H is typically on the order of
lOkG.
Term A is of the same form as the classical dipoledipole inter
action. For a given 6., , spin I. sees a local field due to I, which
either adds to or subtracts from the external field H . Note the
o
dependence on 9., .
jk
Term B is quite interesting. Written in the form,
^ =  i^^^k + ^]A)^^  3 ^°^^^i^\ (2.8)
it is evident that it connects one state of zero energy with the
other; i.e., I. up I down with I up I. down or the inverse. It is
J K K J
also referred to as the flipflop term. The effect of this term which
also commutes with the Zeeman Hamiltonian is to limit the lifetime of
a given zero spin state. First order perturbation theory provides the
mathematical formalism necessary to determine quantitatively its effect
on the zero spin Zeeman state, but a more qualitative statement of the
effect of the finite lifetime on the precision of the zero spin state
may be made using the uncertainty principle. The uncertainty principle
argument is quite interesting and is developed in detail on page 17 of
Andrew (8) .
Terms C and D flip one spin only and thus connect states with both
spins up or down with zero spin states.
Terms E and F flip both spins, thus connecting states of total
spin 1 to states of total spin 1.
In first order, only terms A and B contribute to splitting of the
energy levels. Terms C, D, E, and F do produce a second order energy
shift which is so weak as to be generally ignored if one is concerned
only with the average dipole contribution to the magnetic field at a
12
given nuclear site. These terms are however most important in the
situation where one is interested in molecular motion because of their
strong dependence on the polar angles. They have the effect of causing
transitions among the Zeeman states when the molecule is not station
ary. This problem was first considered in the classic Bloembergen,
;
Purcell, and Pound paper (9).
Terms A and B yield valuable information concerning the local
magnetic field at a nuclear site in a rigid lattice. Van Vleck (10)
has investigated this problem in detail and derived an expression for
calculating the mean square local magnetic field at a given nuclear
site in a rigid lattice xjhen dipoledipole interactions among all
nuclei in the solid are allowed. The effect of rapid, isotropic mo
lecular motion on terms A and B is to average the angular factor to
llj
zero. Theories have been developed to relate the change in the local
field from the rigid lattice value to molecular motion.
2.3.3. The Quadrupole Interaction
To this point we have considered only interactions of a magnetic
nature, with only brief comment in Sec. 2.2. that the nitrogen nucleus
possesses an electric quadrupole moment. The quantization of the
electric quadrupole moment of a nucleus along an electric field gradi
ent in space is analogous to the quantization of the nuclear magnetic
moment along an externally applied magnetic field with one important
^ exception: In diamagnetic materials the term in the Hamiltonian due
to the externally applied magnetic field is the predominant magnetic
term whereas the molecular electric field gradient usually far exceeds '
that which one is able to produce across the sample without breaking
down the dielectric material. The molecular field gradient at the
13
site of a nucleus is generated by the surrounding charges, both elec
tronic and nuclear. Excellent derivations of the electric quadrupole
interaction Hamiltonian are given in Slichter (6) , Abragam (7) , and
Das and Hahn (11) . The field gradient is expressed as a 3 x 3 tensor ■
with nonzero offdiagonal elements when referred to a randomly orient
ed coordinate system, but one may diagonalize this tensor with the
proper choice of molecular coordinate system. By convention the mo
lecular coordinate system is chosen so that the principal components
of the tensor obey the relationship
IV I > V I > IV 1. (2.9)
' zz' — ' yy' — ' xx'
The field gradient, eq, and asymmetry parameter, ri, are defined by
Eqs. (2.10) and (2.11).
eq = V^^ (2.10)
V  V
n= % ^^ (2.11)
zz
Under these conditions the electric quadrupole Hamiltonian, ^^ , is
given by
2
^Q = fefi^) ^'"l  ^' ■" h^'''' ^ ^"')>' <2.12)
where e is a unit electronic charge, I the spin of the nucleus, Q the
quadrupole moment of the nucleus, and I and I are respectively the
raising and lowering operators.
The ammonia molecule possesses C„ symmetry about the V axis,
which is identical to the z axis in Fig. 1. It has been experimentally
determined (12) by Xray analysis that this rotational symmetry is
preserved in solid ammonia. This is further supported by quadrupole'
resonance work (13) in solid ammonia in which a single absorption line
14
was found. Thus n = and Eq. (2.12) reduces for the case of the spin
1 nitrogen nucleus to
K = 4^ {31^  I^}. (2.13)
Q 4 z
A point of interest in this Hamiltonian is that the I^ operator appears
J as a squared term leading to degenerate eigenenergles for the m = +1
states. The factor eq in Eq. (2.13) is dependent on the environment of
the nucleus whereas the nuclear quadrupole moment, Q, is an intrinsic
property of the nucleus itself. It is conventional to define the term
(e qQ/h) as the quadrupole coupling constant expressed in Hz. The
quadrupole coupling coiistant for the case of nitrogen in ammonia is
(14) 4.08 MHz in the gaseous state and varies in the solid state from
3.16 MHz at 77 K to 3.08 MHz at 193 K (approximately 3 K below the
normal melting point) . The quadrupole Hamiltonian is thus approxi
mately equal in magnitude to the Zeeman Hamiltonian and cannot be
treated easily as a perturbation as can jLj*
2.3.4. Magnetic Interactions of Nuclei with Molecular Electrons
The Zeeman Hamiltonian introduced in Sec. 2.3.1. is rigorously
accurate only for the case of completely isolated nuclei. We now
consider the magnetic interactions possible in a diamagnetic molecule
between the nuclei and the moleciilar electrons. The possible inter
actions of this nature are separable into two broad categories:
a) Interactions that are dependent on the magnitude and in some cases
the direction of the externally applied magnetic field, and b) Those
independent of either the magnitude or direction of the applied field.
Interactions of type a) involve a coupling of the molecular elec
trons to the externally applied magnetic field, H^, which has the
)
J
15
indirect result of adding to or subtracting from H at a given nuclear
site. Abragam (7)^ p. 175 states that this effect may arise by two
mechanisms. First the Larmor precession of electronic charges about
H generates at a nuclear site an effective magnetic field H propor
tional to H . Secondly the external magnetic field distorts the elec
tronic shells which adds another component H to the field at a nuclear
P
site. The contribution of H + H , along H may be function of the
p d o
relative orientation of the molecule with H . This effect is usually
very small and observable only in liquids where the only time average
of the component of H, + H along H affects the Zeeman Hamiltonian.
dp o
Thus we can write
or more simply
H H H H^
H = H + ^^ + ^^ (2.14)
H I iH I
' o' ' o'
H = H^(l  6^). (2.15)
The factor 6 is known as the chemical shift and may be either
positive or negative. The experimental fact that it is different for
each molecular species is exploited to tremendous advantage by chemists
in both quantitative and qualitative analysis of unknown substances.
The chemical shift is generally given referred to some solvent, typi
cally tetramethyl silane, and is for protons quite small; e.g., in the
case of NH_, the chemical shift of the protons in NH relative to the
J protons in H2O is only 0.43 parts per million ,(15) . The effect of the
dipolar interaction, x^rhich at H =10 kG causes a shift on the order
of a few hundred parts per million would obviously totally wash out
the effect of the chemical shift were it not for the fact that in
I liquids the dipolar contribution to the magnetic field at a nuclear
J
J
16
site averages to zero due to rapid molecular reorientation.
It is possible to obtain information concerning molecular motion
from this interaction if and only if 6 is a function of the direction
o
of H relative to molecular coordinates. In the case of ammonia no
Information will be obtained from this interaction for reasons that
will soon become apparent.
Interactions of type b) prove to be quite interesting in the study
of ammonia. This interaction was discovered independently by Hahn and
Maxwell (16) and Gutowsky and McCall (17). The paper by Hahn and Max
well is particularly interesting because transient NMR techniques are
used. It was found experimentally that this interaction was independ
ent of sample temperature and applied field and could be represented
by the Hamiltonian,
^iss = ^^^1*^2' <2.16)
where h is Planck's constant, J the indirect spinspin coupling constant
expressed in Hz, and I and I. are the spin operators for two nuclei
on the same molecule. The origin of this interaction, which is known
as the indirect spinspin interaction , is dependent on the details of
the molecular electronic structure. It is sufficient for this work to
consider this interaction as a communication between two nuclei on the
same molecule through interaction with the molecular electron cloud,
i.e., the dipole moment of spin I distorts the electronic structure
of the molecule, and this distortion produces at the site of nucleus I„'
a small magnetic field with which it interacts. It is important to
note that although this is effectively a magnetic dipoledipole inter '
action between nuclei 1 and 2 it does not depend on the relative ori
entation of lab and molecular coordinate frames, and is thus not
17
y
J
averaged out by rapid molecular reorientations. There are however two
circumstances which tend to average the effect of J4, to zero: a) If
ss
the lifetime of a given nuclear Zeeman state of either I^ or I is much
shorter than t = 1/J then the other nucleus 'sees' a time average of
the magnetic field produced at its site due to the rapidly fluctuating
nucleus, which averages to zero over the period t = 1/J, or b) If
either nucleus I^ or I undergoes chemical exchange at a rate, v >> J,
then the exchanging nucleus will 'see' a fluctuating magnetic field due
to the fact that the other nucleus will be in different states on
different molecules.
The indirect spinspin coupling constant has been measured for
ammonia (18,19) to be
Voton  proton = N^l H^
and J ^ ^ = 143.61 Hz.
proton  proton ' I
Obviously this small interaction is totally negligible in the solid
where the dipoledipole interaction energy is typically several hun
dred times this value.
2.4. Summary
Experimentally, we are able to consider the various interactions
with the respective species of nuclei — nitrogen and hydrogen — as to
tally separate experiments because the wide difference in magnitude
of their respective Zeeman Hamiltonians allows us to adjust our ex
perimental apparatus to be sensitive only to small no n over lapping
frequency intervals at the center frequency corresponding to a Zeeman
transition energy. Note that although
0V a»~w m% o i t. h^e S i.«jrT^.at>fvlo^ lM igr>*
y
18
Voton ^ ^^g
V .
nitrogen
the ratio of the respective Zeeman energies is
nitrogen
This is due to the more effective coupling of the proton magnetic mo
ment to the magnetic field. When expressed in frequency units, Hz,
using the Planck relationship
we have.
E , ^ = liv u ^ . (2.17)
photon photon
V ^ = 13.8 V .
proton nitrogen
With this simplification of the problem in mind, we break the experi
ment down into two sections a) information obtainable from experiments
concerned with the nitrogen nucleus and b) information obtainable by
use of the protons as the probe.
To summarize, we list the possible interactions that each molecular
I nuclear species may experience when immersed in an aggregate of identi
cal molecules.
A. The Nitrogen Nucleus
1.) Intramolecular dipoledipole interaction with each
of the three molecular protons.
2.) Quadrupole interaction with the electric field
J gradient at the nitrogen site.
3.) Indirect spinspin interaction with each intra
molecular proton.
4.) Chemical shift interaction.
5.) Intermolecular dipole interaction with both protons
19
and nitrogen nuclei of neighbor molecules.
B. The Hydrogen Nucleus.
1.) Intramolecular dipoledipole interaction among
protons.
j 2.) Intramolecular dipoledipole interaction with
. nitrogen.
\
3.) Indirect spinspin interaction with nitrogen.
4.) Indirect spinspin interaction among intra
molecular protons.
5.) Chemical Shift Interaction.
6.) Intermolecular dipoledipole interaction with
both protons and nitrogen nuclei of neighbor
molecules.
)
CHAPTER III
J INTRODUCTION TO NUCLEAR MAGNETIC RESONANCE
AND RELAXATION
3.1. Thermal Equilibrium of the Nuclear Spin System
As stated in Sec. 2.4., for the case of the nuclear species pre
sent in NH^ we may experimentally consider each nuclear system inde
pendently. Thus we consider a system of N spins/unit volume (spin I)
immersed in a magnetic field, H . Let the spins 'weakly interact' so
as to approach the ambient temperature of the lattice. We attempted
in Chap. II to give some idea of what 'weakly interacting' means in
terms of the molecular Hamiltonian. Ultimately the details of the
interactions must be considered, but for the moment we assume only that
the time scale of the approach of the spin system to equilibrium is
short enough that we are able to observe the effects of thermal equi
librium vs. nonequilibrium in a reasonable amount of time.
Appealing to Boltzmann statistics as outlined by Reif (20) pp. 257
261 the equilibrium nuclear magnetization' is found to be
M = N yfi
^ CTNH
I 2 krl 2 ^^^^ 2kT
(3.1)
V A tremendous simplification of Eq. (3.1) is possible for the usual
magnitudes of applied fields, H < 20 kG, if one works at temperatures
3
above ~ 10 K, i.e., when
^ « 1. (3.2)
20
)
21
one may write for Eq. (3.1):
M N 2 2
^o ^H^= 3kF^ ^^^+1) (3.3)
o
The magnitude of the static nuclear magnetic susceptibility, x ,
o
is typically 10 to 10~ . Therefore, for a laboratory field of
10 kG one must detect a superimposed field of ~ 1 microGauss: A non
trivial experimental problem using static techniques. Static techni
ques are difficult experimentally and not very fruitful when one is
interested in the study of molecular motion, thus other methods must
be considered.
3.2. The Nuclear Magnetic Resonance Phenomenon
We have already established that the effect to be measured is quite
small and is likely to require correspondingly sophisticated experi
mental techniques. Highly elegant techniques have been devised to
detect this very small effect: All involving the use of a transducer
which transforms the nuclear magnetic moment into an electrical sig
nal which is amplified in some cases over 10 times, recorded, and
:em.
finally interpreted in terms of behavior of the nuclear spin systc
Thus our problem becomes one of removing a very small electrical sig
nal from ubiquitous electrical noise, which we define quite generally
as any signal which has no correlation with the nuclear signal. This
problem is taken up in the section on experimental techniques; so for
now we simply state that it is to our great advantage to work with a
signal of a frequency on the order of 10 MHz. Hence, we are faced
with the problem of transforming a static, very small magnetic moment
through a suitable transducer, which must not affect the nuclear spin
system, into an electrical signal with a characteristic frequency
22
around 10 I'lHz.
• Nature works with us in our task by providing the spins with angu
lar momentum as well as a m.agnetic moment. The equilibrium nuclear
spin situation is characterized by not only a total static magnetic
^ moment M but also a total static angular momentum L which is related
JO o
to M by
o •'
\ = ^\' (3.4)
These quantities are of course quantiom statistical expectation values
of quantum mechanical operators and the temporal development of a non
equilibrium situation must be handled quantum mechanically. Although
what follows will appear strictly classical, the motion of the expec
tation value of the nuclear magnetization vector has been rigorously
justified quantum mechanically by I. I. Rabi, et al (21).
We now consider the motion of the spin system described above
given that H^ = H^^ and at t = 0, M = M i. The temporal development
of an angular momentum vector is described by
dL
^ = torque = MxH^ (3.5)
and using Eq. (3.4) we find
dM  
^= yMxH^ . • (3.6)
If we transform to a coordinate system with a common stationary origin
but rotating at an angular velocity oJ relative to that above we find,
) using the relationship on p. 133 of Goldstein (22),
'dM *
M*xy(H^ + Y ) • ' ^^'''^
IdtJ
The star indicates a quantity measured relative to the rotating refer
ence frame. Now if the condition
23
y
Axis of Sample Coil
x"
Fig. 3. THE LABORATORY S STAR COORDINATE SYSTEMS
rf Amp.
V{t) =V.sinu:it
Sample Coil
Fig. 4. THE ROTATING COMPONENTS OF A LINEARLY POLARIZED
OSCILLATORY MAGNETIC FIELD
24
US  bi = ~ yH = the Larmor frequency (3.8)
holds, it is apparent that M is not a function of time. If the two
coordinate systems were aligned at t = 0, M remains aligned with the
o
i axis. Thus we have the situation in which M precesses about H
o '^ o
at an angular velocity given by Eq. (3.8). If we have the same situ
ation except we enclose the sample in a coil of wire, stationary in the
lab frame with its axis normal to H , then a voltage is induced in the
coil due to the rotating nuclear magnetic moment which appears to the
coil as a sinusoidally varying change in magnetic flux with a funda
mental angular frequency given by Eq. (3.8). This coil is the trans
ducer, mentioned earlier, from which we obtain the signal voltage. This
is of course a nonphysical situation as the Boltzmann distribution is
destroyed when M is not colinear with H and the spin system immedl
o o
ately begins to seek equilibrium. Furthermore, we have assumed that
the local field is the same constant value at each nuclear site — also
nonphysical.
We now consider the behavior of the spin system under the influence
of a small, laboratory generated perturbation to the Zeeman Hamiltonian.
Let the perturbation be the application of a voltage,
V = V^sincot, (3.9)
to the coll surrounding the sample. For geometric simplification we
position the coil as shown in Fig. 3, with its axis along the y axis
and geom.etrical center at the origin. This voltage produces a reason
ably homogeneous linearly polarized magnetic field,
«! = ^2H^sincjt, (3.10)
at the sample which can be brokendown into two counter rotating cir
cularly polarized field components as shoxra in Fig. 4. That the
;
y
25
component rotating in a sense opposite to that of the free precession
of the nuclear spin system has negligible effect on the spin system
is demonstrated by Abragam (7), p. 21. We consider only the component
ft
rotating in such a manner that it lies out the x axis. Our initial
conditions are such that att=0,M=M^ and both the lab and ro
o
tating frame are superimposed. These initial conditions are experi
mentally met by first allowing the spin system to achieve equilibrium
in the field H = H k, then turning on E . Eq. (3.7) thus becomes
(f]*=M%[H„+^)t%H^Il. (3.11)
We define an effective field as measured in the * frame by the equa
tion,
H = (H + )k + H,i . (3.12)
e o Y 1 ^ ■^
►*
The angle that H makes with the z axis is
r
= tan
h
(H + oj/y)^
o
(3.13)
Note that 9 = u/l when the resonant condition (Eq. (3.8)) is satisfied.
We now transform to a ** coordinate frame, which rotates at an angular
velocity, ^, relative to the * frame. Eq. (3.12) transforms to
dMi — ** * Q
°''' =M xy[H^+]. (3.14)
dt
Analyzing Eq. (3.14) by analogy with the previous case we see that the
^*
magnetization vector precesses about H when viewed in the * frame.
e
Unless specifically stated, the resonant condition will always
apply in this work. Thus if we let n = yH, i , Eq. (3.14) indicates
that the magnetization vector, as viewed in the * frame, precesses about
the i axis at an angular rate,
n =yH_. (3.15)
)
26
We now define a 11/2 pulse as the application of voltage given by
Eq. (3.9), with ca = w , to the coil for the time, t ,„, necessary to
O TT/Z
precess the magnetization vector dov7n to the x,y plane. From Eq. (3.15)
we see that the pulse duration for a 7t/2 pulse is.
it/2  2^ ^3.16)
Generalizing, the pulse duration for a t, pulse, where ^ is expressed
in radian measure, is
^A = ~l~ • (3.17)
We have implicitly assumed that H is approximately uniform over
the sample. For this simple analysis we have assumed the magnitude of
H^ to be immaterial, but it will soon become apparent that H will
possess certain lower bounds which depend on the sample studied.
Thus, the task outlined in the first paragraph of this section is
complete. To measure the equilibrium magnetization of the spin system:
We simply allow the spin system to come to equilibrium immersed in an
applied magnetic field, apply a v/l pulse to the coil thereby rotating
M down into the x,y plane, then monitor the voltage induced in the
coil by the precessing nuclear magnetization. From the geometry of
the coil and characteristics of the electronics, m [ may be calculated.
3.3. Continuous Wave vs. Transient NME.
3.3.1. General Remarks
The nuclear resonance phenomenon is observed, both in continuous
wave NMR (CWMIR) and transient NIIR (TNilR) , as the response of the
nuclear spin system to an externally applied perturbation. The differ
ence in the two methods arises primarily from differences in magnitude
;
)
27
of the applied perturbation. In the case of CWNMR one continuously
applies a very small rotating H field, a field so small that the ther
mal equilibrium is not appreciably affected. The absorption of energy
by the spin system is then measured as a ftmction of H rotation fre
quency. This absorption curve when normalized gives one the distribu
tion of Larmor frequencies (thus local magnetic fields) over the spin
ensemble. This may be contrasted with the case of TNMR whereby one
applies rotating H fields of such a magnitude that the Boltzmann pop
ulation of levels may be completely destroyed or even inverted. An
extensive variety of TNMR techniques has evolved in the preceding two
decades, but we limit our discussion to the technique characterized by
the following two distinct experimental steps :
1) A highly intense perturbation is applied to the spin system,
viz. a 90° pulse.
2) The response of the spin system to the perturbation is then
observed without further perturbation .
3.3.2. Relationship Between the Lineshape and the Free Induction Decay
In this discussion we assume that a perfectly uniform external
field H = H k is available, and that the rotating field H, is per
o o 1
fectly uniform over the sample. Furthermore, we assume that no instru
mental effects are Introduced. These effects are to be considered in
connection with the experimental apparatus and neglecting them for now
simplifies the discussion.
Perturbations of the Zeeman Hamiltonian, as outlined in Chap. II,
produce a distribution g(H) in the z component of the magnetic field
as seen at the sites of the nuclei. Let g(H) be a normalized
J
28
distribution function centered about H with g(H)dH defined as the
fraction of nuclei/unit volume which 'see' a magnetic field between
H and H + dH Gauss. The normalization condition is:
OS
/g(H)dH = 1 . (3.18)
—00
Specifically we will consider a sample of protons in thermal equi
librium immersed in an external field, "b. . Denote the thermal equi
librium nuclear magnetization as M = M k. The co5rdlnate systems
o o ■'
and initial conditions are the same as considered in Sec. 3.2. A 90°
pulse of radiofrequency (RF) voltage at frequency, oj , is applied to
the coil. Note that only the fraction of spins g(H )dH 'see' an effec
txve field H^ = H^i . The fraction of spins g(H +AH)dH 'see' an effec
tive field
t* Aft A*
H^ = AHk + H^i , (3.19)
and will precess about this field for the duration of the pulse. Thus
the condition that must be met experimentally for a 90° pulse to have
any significance for the complete spin system is
h^ » 6H/2, (3.20)
where 5H is roughly the width of the distribution function g(H). In
_ft ^*
this case H  H^i and essentially all the spins precess as a group
A* A*
about H^i down to the +yi axis. This condition is sometimes not
easily met experimentally when the sample is solid due to the very
large dipole coupling sometimes present, but in liquids it is usually
quite easy to satisfy. For this discussion we assume this condition
is met.
Fig. 5 (a. ) depicts the situation as seen in the stationary labora
tory frame immediately following initiation of the 90° pulse. Fig. 5(b.)
29
(a.)
J
\
A* kit," "At
(c.)
(e.)
(d)
M.
(f.)
X'
T,>Ti<i:
i.'^^^r,
(A,* = M.V
iti;>T,
Fig. 5. 'ideal' SPIN SYSTEM BEHAVIOR FOLLOWING A 90° PULSE
Mw ■^"•^*s«*sn»^.
J
30
depicts the motion of M as seen in the * frame. Under the influence
o
of a 90 pulse, M precesses about the x axis, from the z to the y
axis. The remaining parts of Fig. 5 depict the reestablishment of
thermal equilibrium.
The change in the value of M (t=0) = M i for time, t>0, is brought
on by two processes, both of which are a manifestation of spin ensem
ble migration to thermal equilibrium. We note that statistical ther
mal equilibrium requires not only that a Boltzmann population exist
among the Zeeman states, but also that the spin state fxmction phase
factors are randomly distributed (random phase hypothesis). A 90°
pulse has the effect of equalizing the spin state popxilations and cre
ating a nonrandom distribution of spin state function phase factors.
The distribution of phase factors is such that at completion of the
pulse the vector sum of the nuclear magnetization/unit volume is
M(o) = M J . The tv/o processes leading to ultimate destruction of
M *(t) are: The recreation of a Boltzmann distribution, and the re
establishment of the random phase distribution. The effect of both
processes is implicitly contained in the lineshape function, f(aj).
The normalized lineshape fimction may be viewed as a precession fre
quency distribution function, where f(w)dco is the fraction of spins
with Larmor precession frequency measured in the * frame to be between
CO and to + doj. fCu) is symmetric about oj = o. H ^(t) will thus be
destroyed both by processes that tend to fan the spins out and proc
esses that induce Zeeman transitions. At any time, x, M ^(t) can be
written as
M *(t) = h I cosuxdN^d), (3.21)
CO
y
31
where dN (t) = The number of spins/unit volume with precession fre
quency in the * frame between co and co + do), and y = magnetic moment/
spin. Using the fact that f(to)dco represents the fraction of spins with
precession frequency between co and co + du, the above sum can be written
as
My*(t) = M^G(t), (3.22)
where
CO
G(t) = /f (co)cosojtdco. (3.23)
.OS
Eq. (3.22) contains all the physical information available concerning
the distribution of Larmor frequencies contained in the spin ensemble.
It is appropriate at this point to introduce the phenomenological
equations of Bloch (23) . This set of linear coupled differential equa
tions describes the temporal behavior of the nuclear magnetization.
The theory has been quantum mechanically verified under certain condi
tions by Wangsness and Bloch (24) and gives particularly good quantita
tive information in the case of liquids. Written in vector notation,
Bloch' 8 equations are:
jrj Mi + Mj MM
 = Y^  ^ ^  V°^^ • (3.24)
2 1
Bloch 's equations, for the case of free precession following a 90°
pulse lead to a Lorentz lineshape function:
and the halfwidth at half intensity is
A = ^. (3.26)
The constant T^ is called the spin la ttice relaxation time — the charac
teristic time constant describing the motion of a nonequilibrium Zee
32
man level population toward thermal equilibrium with the lattice. T^
is the spinspin relaxation time — the characteristic time constant de
scribing loss of transverse magnetization. Note that T„ implicitly
contains any mechanism which causes loss of transverse magnetization;
J this includes processes governed by T . Thus the lineshape function
I as derived from Bloch's equations implicitly contains the effect of the
finite lifetime of a Zeeman state. We now consider the measurement of
My*(t).
Recall that the voltage V(t) induced in the coil, which lies with
its axis out the y axis of our laboratory frame, is proportional to the
first time derivative of M (t). Transforming Eq. (3.22) to the sta
tionary frame we find
M (t) = M^G(t) s±n(i^^t  90°). (3.27)
In taking the time derivative of Eq. (3.27) we recognize that Eq. (3.22)
is a very slowly varying function of time, thus its contribution to
V(t) is negligible. Eq. (3.22) represents the 'information' we wish
to extract from the 'carrier', sinco t. Performing the indicated oper
ations we find
V(t) = aM G(t) 10 sinto t, (3.28)
. o o o
i
where a is a proportionality constant. This voltage might appear on
an oscilloscope as shown in Fig. 6(b.), depending on the form of G(t).
In practice, however, V(t) is 'detected' and only the 'information',
I Eq. (3.22), appears on the oscilloscope screen (see Fig. 6(c.)). We
assume that phase coherent detection is used so that both positive and
! negative excursions of M !;(t) are observable. The trace V, (t) is
known as the free induction decay , henceforth to be referred to as the
FID. All the constants of proportionality in Eq. (3.28) are lumped
33
ng.S(a). TEMPORAL DEVELOPMENT OF M^t) FOLLOWING A 90° PULSE
V(t)
Larmor Frequency rf
nformation Envelope
ISO'rf Phase Shift at Crossover
Fig. 6(b.). VOLTAGE INDUCED IN SAMPLE COIL
Fig. 6(c.). PHASE COHERENT DETECTED SAMPLE COIL VOLTAGE
34
into the constant, a, and we have,
V^(t) = aG(t) (3.29)
We examine G(t) in more detail at this point. Using the fact that
f(a)) is an even function of w and that w = 2itv , we write,
CO oo
G(t) = /f(v)cos 27rvt 2TTdv = 2Tr/f (v)EXP(2iTiv)dv. (3.30)
—CO — CX)
Thus G(t) is proportional to the Fourier transform of the Larmor dis
tribution function for the spins. Now we rearrange Eq. (3.29) and
multiply through by EXP(ia)'t).
EXP(ia)'t) ff (ta)EXP(iiot)daj = V rt)EXP(iaj't) (3.31)
ad
— 00
Integrating over all tspace and assuming the order of integration is
interchangeable, we find.
/duf(uj) /EXP[it(ajoj')]dt =  /V rt)EXP(ito't)dt. (3.32)
_, a ' d
•mOO —CO —CO
Using the wellknown integral representation for the 6function,
eo
/EXPit(a)a)')]dt = 2^6(waj'), (3.33)
—00
and performing the integration over oj in Eq. (332), we have,
00
f (o)') = V^ /v^(t)EXP(iu)'t)dt . (3.34)
ZTTa ' d
—CO
Thus we have demonstrated that the FID and the distribution function
are related through the Fourier transform.
3.3.3. Moments of the Lineshape
We consider the symmetric normalized lineshape function, f(w),
35
previously defined. The n moment of f (oj) is defined by
00
M^ = /a)'^f(co)da). (3.35)
—CO
Note that all odd moments vanish as a result of the even parity of
f(a)). The importance of determining the moments of a lineshape is
twofold: a) Knowledge of the various lineshape moments provides de
tailed shape information; b) Van Vleck's (10) classic calculation re
veals the equality of the mean square local field at a nuclear site
and the second moment of the lineshape function.
By using the following equation, one may calculate the second mo
ment at the site of a given nuclear spin I due to dipoledipole inter
actions with neighboring I and S spins. This equation is valid only
if all spins are in equivalent lattice positions.
2 3yVi(I+1) (1  3cos^9.,)^
<AH > = —  — ; y 1^^
4 ^ 6
sum over r . ,
I species
2 2
+ hX^(S+l) I , J^ > (3.36)
sum over r.,
S species
where r are the respective intranuclear distances. This calculation
is quite tedious for nuclear configurations of little symmetry. Great
simplifications in this problem are usually possible however thanks to
—6
the rapid r decay of the squared dipoledipole interaction. One may
usually truncate the sum after nearest or next nearest neighbors. An
other simplification occurs for the case of a powdered sample, for
which Eq. (3.36) averaged over all angles becomes.
36
/
<AH^> =I(I+1)y^^ I r~^ + ^ys(S+l) I r'^ . (3.37)
Sinn over sum over
I species S species
Experimentally the n moment is usually obtained by CWNMR line
j shapes, but it may also in some cases be conveniently obtained by
transient techniques. The expression from Abragam (7) relating the
n moment to G(t) (Sq. (3.30)) is:
M„ = (i)nfd!!Itl
l
G(0). (3.38)
t=0
dt
This method of determining M is used somewhat in the literature, but
n '
suffers from a quite critical experimental difficulty: G(t) is ob
scured at t=0 by the finite recovery time of the apparatus. New exper
imental techniques currently in use remove this problem using the tech
nique of solid echos (25,26,27,28) to reflect the t=0 shape to an ob
servable point in time.
The important point here is the additional support of the intimate
relationship between the lineshape and the FID. In the next section,
we hope to give some indication of what may be expected for the FID
of protons in solid NH^.
3
3.4. The Effect of Molecular Motion on the
Free Induction Decay and Lineshape in Ammonia
We begin with the statement of five theorems from Fourier trans
form analysis. The proofs to these theorems as well as an excellent
introduction to Fourier transform analysis are presented in R. Brace
well's (29) text.
Theorem 1 : The Addition Theorem
If f(v) and g(v) have the Fourier transforms F(t) and G(t), re
J
)
37
spectively, then f(v) + g(v) has the Fourier transform F(t) + G(t).
Theorem 2 : The Shift Theorem
If f(v) has the Fourier transform F(t), then f(v + v ) has the
o
Fourxer transform e " F(t).
Theorem 3 : The Convolution Theorem
If f(v) and g(v) have Fourier transforms F(t) and G(t), respective
ly, then the convolution of f(v) with g(v);
00
f(v)*g(v) = /f(x)g(v  x)dx, (3.39)
—CO
has the Fourier transform F(t)G(t).
Theorem 4 : The Autocorrelation Theorem
If f(v) has the Fourier transform F(t), then its autocorrelation
function,
00
/f*(u)f(u + v)du, (3.40)
—OS
2
has the Fourier transform F(t)[ .
We now define the equivalent width of a function W[f(v)],
CO
/f(v)dv
W[f(v)] = """f^^) , (3.41)
if and only if f (o) ?^ 0. W corresponds to the width of a rectangle
having height f(o) and area equal to that under f(v).
Theorem 5 : E quivalent Width Theorem
If a function f(v) has a Fourier transform F(t) and f(o) ?^ 7^ F(t),
then
W[f(v)] W[F(t)] = 1. (3.42)
3.4.1. The FID and Lineshape in Solid NH
We consider only the magnetic resonance of protons in solid NH..
The nitrogen magnetic resonance is weak and smeared out so far as to
y
38
be unobservable in our polycrystalline sample immersed in a 9 KG mag
netic field.
One finds that the lineshape obtained from solids at very low tem
peratures is, in many cases, satisfactorily represented by a Gaussian
shape function. The normalized Gaussian curve is given by
1  2
f (oj) = —  EXP \ , (3.43)
A/2tt 2A
where A is the halfwidth between points of maximum slope of f(a)). We
find the associated FID by substitution of Eq. (3.43) into Eq. (3.29),
i.e.
V^(t) = B e"*^ ^ /^ (3.44)
where B is a constant. The value of T^ is taken to be that value of
time for which V^(t) equals 1/e its value at t=0, i.e.
/2
T2 = t=~. (3.45)
When the temperature of a solid is increased to the point where
motion is very rapid, the Lorentzian lineshape is sometimes a better
approximation to the experimental curve. Eq. (3.29) gives the follow
ing relationship for the FID when the Lorentzian lineshape function
(Eq. (3.25)) is used:
V^(t) = Ae^/^2^ (3,^g)
where A is a constant and T^ is defined in Eq. (3.26) as the halfwidth
at halfmaximum of the normalized Lorentzian lineshape function.
Fig. 7(a) depicts a comparison of the FID predicted by Eqs. (3.44)
and (3.46) with the experimental value measured for protons in NH at
116 K. Fig. 7(b) illustrates f(!:ij) obtained by the method of a^JNl■'IR.
The theoretical curve was empirically fit as well as possible to the
39
— LORENTZIAN
— GAUSSIAN
EXPERIMENTAL POINTS
8 10 12 14 16 18 20 22 24
TIME (MICROSECONDS)
Fig. 7(a.), FID (PROTONS OF NH, AT II6K)
4.2
I90K
UJ
o
iLl
>
cr
o
UJ
N
<
q:
o
z
III K
.•••
■ f  r
60 50 40 30 20 10 10 20 30 40 50 60
FREQUENCY (KHz)
Fig. 7(b.). SOLID NH3 LINESHAPE FUNCTION AT THREE TEMPERATURES
40
experimental curve. This comparison is for the sake of illustration
only. Quantitative results on lineshapes from TNMR when a FID is as
short as that for protons in NH_, although perfectly feasible, require
techniques and equipment not used in this laboratory.
We now consider, qualitatively, the effect of molecular motion on
a lineshape. Suppose that the FID has been measured at a temperature
low enough that all motion is essentially of a ground state nature.
The f (to) measured under these conditions is called the rigid lattice
distribution function and the width of the distribution between points
of maximum slope will be referred to as the rigid lattice linewidth .
The spinspin relaxation time corresponding to this distribution func
tion will be symbolized by T . T sets the time scale in the low
temperature region: Any perturbation which has a characteristic fre
quency less than approximately T„ cannot affect the linewidth because
the loss of phase coherence among the spins is approximately complete
before the perturbation can produce any change in the magnetic envi
ronment of a given nucleus. To elucidate, we characterize the behavior
of the precession frequency to(t) of a given spin as seen in the * frame
by the following parameters:
1 ''''
a.) <co> = — / oj(t)dt = average precessional frequency (3.47)
uj 0) o
over the time period t .
b.) T = the average lifetime of a welldefined nuclear
precessional state.
c.) CO = the root mean square deviations from <to>
rms To
For the low temperature case x >> T. and o is quite small.
CO 2r rms ^
On the average, precessional states are thus welldefined for many
41
times the characteristic time for the complete spin system to lose
phase coherence. The upper bound on the average lifetime of a pre
cessional state for a given nucleus is T , as previously defined. Each
possible molecular motion can be characterized by the magnitude of the
fluctuations from <a)> that it is capable of producing at the nuclear
sites, i.e., we expect different values of co for different types of
rms ■"
molecular motion. As the lattice temperature is increased, co is
rms
determined almost exclusively by the type of molecular motion activated.
The average lifetime t decreases with increasing temperature because
the probability of occurrence of a given mode of molecular motion in
creases. In many cases the rate of occurrence of a given mode of motion
may be represented by a welldefined mathematical expression. The form
of this expression will be considered in Chap. IV when its effect can
be explained using data obtained as an example.
As T decreases (temperature increases) <cj> is not appreciably
affected until t  "Y. because, as mentioned previously, the important
time interval is T„ . When the point x == T„ is reached the molecular
2r cj 2r
motion becomes effective as a line narrowing mechanism and a linewidth
reduction occurs. The magnitude of the linewidth reduction depends on
the details of the motion involved, i.e., the magnitude of the average
fluctuation it produces. The reasoning here is that very large ran
domly fluctuating precession states tend to average <a)> to zero much
quicker than a smaller random fluctuation. The bulk effect is that the
average of <aj> over the spin ensemble decreases. The rate of fluc
tuatlon due to a given mode of molecular motion is initially an ex
tremely rapid function of temperature which tends to level out, ap
proaching a limiting value as temperature increases (see Fig. 17). The
42
sharpness of the linewidth transition is governad by the temperature
dependence of the rate of fluctuation when the condition x  T„
is satisfied. As t decreases beyond T , little additional decrease
w 2r
in the linewidth occurs due to this process and one is able to define
another relatively temperature independent T' and t',
^ 2 w
Many substances exhibit more than one fairly sharp linewidth transi
tion in the solid region between the region of rigid lattice tempera
tures and that of the solidliquid transition. This effect is due to
the thermal activation of additional modes of molecular reorientation
which contribute to the averaging process and further narrow the line.
These additional linewidth transitions occur when the lifetime for the
prevailing average precessional state becomes approximately equal to
the prevailing spinspin relaxation time.
We have considered in the previous paragraphs the harrowing of the
normalized lineshape, but have not mentioned the effect of this mole
cular motion on the shape of the curve. This is not a trivial problem
to consider quantitatively, but a very general qualitative statement
of how the distribution changes may be made. Molecular motion has the
effect of producing fluctuations in the local magnetic field at nu
clear sites. Fluctuations may be placed in one of the two following
rather vague categories: a.) Fluctuations which have a peak value
very close to the rigid lattice root mean square local field, and
b.) Fluctuations which have a peak value much greater than the rigid
lattice root mean square local field.
Fluctuations of type a.) are produced by motional processes which
are not capable of causing much change from the rigid lattice inter
nuclear distances. The averaging process in this case is an average
43
over angular factors appearing in the dipoledipole Hamiltonian. One
observes a line narrowing due to the averaging process previously con
sidered, but little or no change of character from an approximate
Gaussian type shape function is observed. An example of this type of
narrowing occurs in the case of solid NH when the molecules undergo
hindered rotation about the C symmetry axis. Note that any intra
molecular interaction causes fluctuations of this type when the mole
cule undergoes molecular reorientation.
Fluctuations of type b.) are caused by processes capable of pro
ducing large changes from the rigid lattice interproton distances;
e.g., 1.) molecular collisions in liquids, or 2.) molecular self
diffusion in solids via the formation of Frenkel type lattice defects,
or 3.) molecular reorientation in solids which involves a large re
duction in internuclear distances, e.g. isotropic molecular reorienta
tion in solid NH„. The large fluctuations in the local magnetic fields
at nuclear sites produced by these processes tend to increase the prob
ability of high energy Zeeman transitions. This leads to a lineshape
with more intensity in the wings than appears in the Gaussian function.
The Lorentzian shape function is a much better approximation to the
physically observed lineshape when the lines are narrowed by fluctua
tions of the type b.).
We need to further emphasize that the previous discussion is of a
highly qualitative nature and quite incomplete. The intention was
that of only providing some idea of how molecular motion alters the
experimentally o bservable lineshapes .
44
3.4.2 The FID and Linesh a pe in Liquid NH »
Pople (30) has considered in detail the effect of N quadrupole
relaxation on the proton multiplet lineshape in molecules such as NH
14
If N had a Zeeman lifetime longer than that for the protons, f(a))
could be well approximated by a sum of three delta functions . How
ever, with fluctuations of the nitrogen nucleus taken into account,
Pople finds that the protons 'see' an average J which has the effect,
for moderately slow quadrupole relaxation, of simply broadening the
6function lines and producing three Lorentzian lines. The two satel
lite lines are broader than the central line, the ratio of halfwidths
at half maximum being,
J : ^ : ~= 3:2:3. (3.48)
2 2 2
Consider the definition of the convolution f(v)*g(v), defined in
Theorem III, with f(v) a Lorentzian Lineshape (Eq. (3.25)) and
g(v) = 6(v  J).
f(v)*g(v) = /f(T)6(vJT)dT = f(vJ) (3.49)
The effect of convoluting 5 (v J) with f (v) is thus simply to shift
the origin of f(v) from v=0 to v = J.
We may thus represent Pople 's results, to fairly good approximation,
when T2J > 1 as
f (v) = 3
f(T,,v)*6(vJ) + f(T' v)*5(v+J) + f(T.,v)*5(v=0)
, (3.50)
where — : rrr : Tf satisfies Eq. (3.48), and
2 2 2
T_2 dv_
— ^ (1/T„)^ + (2ttv)'
/f(T„,v)dv = /^ :^  = i (3.51)
2' i TT /, ,^ ^2 , ,^_,^2 3
2>
45
Using the distributive property of the convolution, we find
f (V) = J
f(T2.,v)*[5(vJ) + 5(xHJ)] + f(T^,v)*5(v)
(3.52)
Substituting into Eq. (3.29) and using Theorems 1 and 2 we obtain
V^(t) = C
2EXP(t/T )cos2TTjt + EXP(t/T')
(3.53)
1 2 1
where C is a constant. Since — — = T ^jTTj we may write Eq. (3.52) as
2 2
V^(t) = C{EXP(2t/3T2)[2EXP(t/3T2)cos2TTJt + 1]}.
(3.54)
Recall from Eq. (3.48) that (l/T^) is the halfwidth at halfmaxi
mum of one of the outside multiplets. J is the splitting from v=0
expressed in Hz. Fig. 8 (a.) depicts these relationships.
It is interesting to consider the bracketed part of Eq. (4.54)
plotted as a function of t for various values of
1 1
k =
T2J 43.6T2'
(3.55)
Fig. 8(b.) depicts
V'(t) = V EXP(^x43.6kt)
o J
2EXP
43.6kt
cos(2TTx43.6t) + 1
, (3.56)
the numerical values of which were calculated with the Hewlett Packard
9100A digital computer for k=0.1 and 1.0.
The purpose of this treatment is to illustrate the form of the FID
under this particular set of circumstances. Although qualitatively
correct for k<l, it is completely wrong for k>>l, because the theo
retical prediction that the three lines coalese into one narrow Lor
entzian, for rapid fluctuations of the nitrogen nucleus, is not taken
into account. The proper treatment would predict, for increasing k,
the gradual loss of FID beat pattern and lengthening of T^.
Transient techniques can in principle resolve J splittings on the
46
Fig. 8{a.). LIQUID AMMONIA LINESHAPE AT 201 K (FROM REF.(I9))
t \
k =0.1
k = 1.0
TIME (msec)
Fig. 8(b.). FID EXPECTED FROM LIQUID AMMONIA (PROTONS)
47
order of
J ^^ . (3.57)
2
Thus if ^2^ 1 second, as is the case for many pure molecular liquids,
TNMR can in principle resolve J^lHz. In practice one is limited,
without the use of special high resolution techniques such as spinning
the sample in the field, by the apparent decrease in T„ caused by mo
lecular self diffusion through an inhomogeneous magnetic field.
3.5. The SpinLattice Relaxation Time
In Sec. 3.1. an expression was presented for the equilibrium nu
clear magnetization. The temperature T appearing in this expression
was assumed to be that of the 'lattice' in which the spins reside. We
implicitly assumed that "enough time" had passed, from the time the
sample was placed in the external field to the time M would be meas
o
ured, for the spin system to achieve a Boltzmann population distribu
tion at the temperature of the 'lattice. ' This requires an exchange
of energy between the spin system and the 'lattice' and implies an
interaction between the nuclear spins and the 'lattice.* The spinlattice
relaxation time T, introduced in Sect. 3.3. provides one with a quanti
tative idea of how long one should wait for the spin system to reach
thermal equilibrium, viz. approximately five times T^ .
We introduced T in Sect, 3.3. as a parameter in the Bloch equa
tions. We now investigate the relation of T^ and the Zeeman transition
probability and obtain an expression for T^ in terms of the component
of nuclear magnetization aligned with the applied field. Consider the
case of N spin 1/2 nuclei per unit volume immersed in a magnetic
field H = H k. We assume that the heat capacity of the 'lattice,'
48
housing the spin system, is much greater than that of the spin system.
Let W(+) be the transition probability per unit time from a spin
aligned to spin antialigned state and W(+) the inverse. Let N(+) and
N() be the populations of the +^and j)> states respectively. At
equilibrium the condition,
N(+)W(+) = N()W(+), (3.58)
must hold. The spin system obeys Boltzmann statistics; thus the ratio
of Zeeman level populations is
(+) ^E(+)/kT [+t^H^ + (+7ftH^)]/kT
N ^
()
e
N() E()/kT ^ ' ^^^^^
where T is the lattice temperature. Thus it follows that
W(+) = w(+)e+^o/'^T . We^«o/kT. ^3^^^^
The z component of the nuclear magnetization, M , is thus
M^ = u[N(+)  N()]. (3.61)
The equation for the temporal development of M is obtained from
z
Eq. (3.61), i.e.
dM
^ = 2p[N(+)W(+)  N()W(+)]. (3.62)
dt
Using Eq. (3.59), we find
dM
= 2Wy[N(+)  N()e^^o''^^]. (3.63)
dt
In the case of protons in a 10 kG magnetic field, the exponent in
Eq. (3.59) has the value,
f^^ 3.2 X 10"^
kT T '
Therefore the condition,
yhH
— « 1, (3.64)
49
is satisfied if T > 0.01 K and the exponential function in Eq. (3.63)
can be approximated very well as the first two terms in its series ex
pansion. Using this approximation, Eq. (3.63) may be writt
en:
dM
f^2ii
But for I = 1/2, Eq. (3.3) yields
H N 2^2
= _2_o XJ_
kT 4 •
From Eqs. (3.59) and (3.64) we see that the populations of the two
energy levels at thermal equilibrium are very nearly equal. Using the
(3.65)
(3.66)
fact that
N
N() ^ f ,
and Eq. (3.66), we find that Eq. (3.65) reduces to
dM
T~ = 2W(M  M ).
at z o
(3.67)
(3.68)
Eq. (3.68) has the solution,
M (t) = M
z o
i^Sr^^^^<'/V
where we let
= 2W.
(3.69
(3.70)
Eq. (3.69) although derived by using a number of assumptions and
approximations, nevertheless provides an excellent working approxima
tion for determining T^, even in solids. The most fundamental assump
tion used in the derivation of Eq. (3.69) is that which permitted the
use of statistical techniques, i.e. that between any two spins there
exists a spinspin interaction which is weak compared to the Zeeman
interaction, and on the average is temporally indistinguishable from
any other possible spinspin interaction. This assumption is rigor
ously satisfied for the magnetic dipoledipole interaction between
50
spins if H^ is much larger than the local magnetic field and all spins
undergo totally uncorrelated motion. Obviously this is not true for
the case of more than one identical spin belonging to a rigid molecule
because the motion of these spins is clearly correlated. A great deal
of work has been done to determine the effects of correlated spin mo
tion on the temporal path of M^ from a nonequilibrium to an equilib
rium situation. Including these effects leads to an equation much of
the same form as Eq. (3.69) but with the single exponential decay re
placed by a sum of decaying exponentials with different characteristic
decay constants and different weighting factors. Eq. (3.69) is, how
ever, a very good approximation even in the extreme case of intramo
lecular proton relaxation in ammonia via molecular reorientation.
Another complication to the theory occurs when one considers nuclei
of spin I > 1/2. This case also leads in general to a solution for
M^(t) consisting of a weighted sum of exponentials with different time
constants. This case will not concern us.
CHAPTER rv
INTERPRETATION OF EXPERIMENTAL RESULTS
4.1. Outline of Experimental Results
We begin with an outline of the actual physical measurements made,
with brief mention of the method used. A thorough description of ex
perimental apparatus and procedure follows in Chap. V.
I. The Solid State of Ammonia
A. Measurement of T vs. Temperature for Protons
1. Region of temperatures studied  65 to 195 K
2. Magnetic f ield 
a.) H = 2.56 kG
o
b.) H = 3.87 kG
o
c) H = 4.88 kG
o
d.) H = 8.22 kG
o
3. Method of measurement  90T90 pulse sequence
B. Measurement of Linewidth of CWNMR Absorption Curve
1. Region of temperatures studied  1 to 195 K
2. Magnetic Field  1.7 kG
3. Method of measurement  CWNMR using magnetic
modulation
C. Calculation of Second Moment of C1JNMR Lineshapes
1. Region of temperatures studied  '1 to 195 K
2. Magnetic field  1.7 kG
51 •
)
52
3. Method  Calculated from CWNMR lineshapes
obtained from previous experiment
II. The Liquid State of Ammonia
A. Measurement of T^^ vs. Temperature for "'"'^N Zeeman
Relaxation
1. Region of Temperatures studied  195 to 239 K
2. Magnetic field  H = 8.9 kG
o
3. Method of measurement  90T90 pulse sequence
B. Measurement of T, vs. Temperature for Protons
1. Region of Temperatures studied  195 to 239 K
2. Magnetic field
a.) H = 0.775 kG
b.) H = 4.88 kG
o
3. Method of Measurement  90T90 pulse sequence
C. Measurement of T^ vs. Temperature for Protons
j
1. Region of Temperatures studied  195 to 230 K
, 2. Magnetic field  4.88 kG
3. Method of measurement  CarrPurcell pulse
sequence
■^ • 2 . Germane Properties of Liquid and Solid Ammonia
! All work reported herein was performed at an almost constant pres
\ sure over the sample of ~ 20 Ib/in^ absolute. The normal melting and
; boiling points of NH^ (1) are respectively 195.36 and 239.76 K. The
anomalously high boiling point for a member of the Group V hydrides
is attributed to the strong hydrogen bonding characteristics of the
molecule.
53 .
The solid phase crystal structure of m^ and ND^ has been deter
mined using single crystal and powder Xray techniques by Olovsson and
I . Templeton (0T) (12) . Measurements were made at temperatures of 77
and 171 K. The study revealed the crystal space group to be P2 3.
. The structure is simple cubic with a basis of four molecules, but may
{ be considered to be a slightly distorted fee lattice with each molecule
I having 6 nearest neighbors, with which it participates in hydrogen
bonding, and 6 slightly removed next nearest neighbors. Fig. 9 depicts
: the normal solid ammonia lattice structure. The view is that looking
down the trigonal axis toward the origin of a cubic cell. The nitro
gen nucleus at the origin does not lie exactly on the Bravis lattice
point, but 0.0401 1 out the trigonal axis of the cubic cell (out of
the paper surface). The important point to notice for an understand
ing of our work is: Each molecule is hydrogen bonded to six nearest
, neighborsall molecules are in equivalent positions. The distance
between nearest neighbor and next nearest neighbor nitrogen atoms in
I NH3 was found to be respectively 3.38 and 3.96 £ with the sample at
171 K. Each molecular free electron pair was interpreted to form
three hydrogen bonds. A bond is formed with one hydrogen of each of
three of the nearest neighbors. Each of the three molecular hydrogens
forms one third of the same type bond with each of the remaining three
I nearest neighbors. A difference synthesis was also attempted to re
J veal the hydrogen positions. The HNH angle (107°) was found to be
I within experimental error of the free molecule value, but the NH bond
I distance found (1.13 X) is 12% larger. This bond length is question
able because of its unusually large deviation from the free molecule
value.
54
)
Fig. 9. The structure of solid NH2 , as viewed toward the origin, down
the Co axis [111]. Hydrogen bonds are indicated by dashed lines. Mol
ecules 1,2, 3, 4, 5, and 6 are each hydrogen bonded to molecule 0. The pro
ton second moment (Sec. 4.4.1) was calculated for the origin molecule
proton marked 0; all contributing protons are marked with x. This dia
gram is based on work reported in ref. (33).
■'^'■ ' i* ". ■ 1 '^ . ■ j rf>^ M »» V,rt .'nW^^ll M M T> » t ,4Wl»»i— '^.•»Jst.«\wUj
55
The existence of two metastable noncubic forms of solid NH3 has
been reported in the literature (31,32). The metastable forms occurred
when NH3 was deposited from the gas phase onto a surface which was kept
at approximately 77K. Our experiment was performed in such a manner as
to exclude any possibility of forming a metastable phase,
A complimentary lattice structure study of polycrystalline ND3 was
made by Reed and Harris (RH) (33) using neutron diffraction techniques.
Neutron diffraction studies give average positions of the nuclei whereas
Xray diffraction reveals m.axima in the electron cloud distribution. We
expect the neutron diffraction m.easurements to be of more use because
it is the average internuclear distance that is associated with NMR
measurements. The (RH) values of NN distance are within experimental
error of the (0~T) measurements on ND^, but the ND bond length of
1.005 ± 0.023 R is much closer to the free molecule value of 1.008
* 0.004 g (1,34) than the 1.12 g found by (0T) . The DND bond angle
of 110.4° ^ 2.0° is however significantly different from both the (0T)
value of 107° and the free molecule value of 107. 4°* 0.2° (34). The devi
ation of a hydrogen bonded D from an NN line was found to be 11.3°
± 1.7°.
Xray powder work has been done on I\fH3 to 4.2K (32) and no crys
tal structure change from the high temperature structure is observed.
Heat capacity measurements from 15K to the vaporization point (35)
reveal a sharp transition only at the melting point. The heat ca
pacity data do display subtle inflections .in the curve at approxi
mately 50 and 165 K. These inflections are interpreted to mark the
onset of different modes of molecular motion, a presumption supported
by this work. Thermal conductivity of solid NH3 in the temperature
iit«=£;«iri:*i«a^jF?=* '>'=»^4«*.*.i«tMp:i.'2bi^A.
56
range 23 to 114 K (36) gives no iuformation concerning molecular re
orientation, but the thermal expansion data of Manzhelii and Tolkachev
(37) in the temperature region 24 to 175 K are interpreted taking into
account hindered rotation of the molecules. The graph of the coef
ficient of linear expansion vs. temperature, presented in (37), ex
hibits inflection points at 50 and 130 K. These inflection points are
interpreted by the authors as being due to hindered rotation of the
molecules.
Reding and Hornig (38) have measured the infrared absorption spec
tra of NH^ and ND^ at 83 K from 300 to 2000 cm""'. That they find no
indication of molecular inversion is quite important in the interpreta
tion of the results of this work. Hydrogen bonding of the molecules
in the solid, they assert, raises the barrier to inversion by several
kilocalories per mole. Torsional vibrational modes which are assigned
to molecular torsional oscillations about axes normal to the molecular
symirietry axis occur at 527 cm""'" and 362 cm""" in NH . A strong absorp
tion line was observed at 261 cm" , but not interpreted.
A quite interesting and important experiment performed by Lehrer
and O'Konski (14) is the N quadrupole resonance work in solid NH ,
NDH^, ND^H, and ND^. The quadrupole coupling constants were measured
as a function of temperature from 77 K to the melting point and the
'quasistatic' value was then calculated to be (3.47 MHz) in both
m^ and ND^, It is pointed out that torsional vibrations about the
molecular syimnetry axis, which is also the V^ axis of the field gradient
c
tensor, are not effective in reducing the quadrupole coupling constant;
therefore only torsional vibrations about axes normal to the symmetry
axis are considered. Bayer theory in conjunction with Reding and
wITll, . 1— >»**> >« r "3VT i .i«^j&*.».«lH
^
57
Hornig's (38) values of vibrational frequency gives an excellent fit to
the experimental data. We interpret these results to mean that little
reorientation of the NH^ molecule in the solid occurs about axes nor
mal to the symmetry axis. Perhaps we should clarify the distinction
between molecular reorientation and torsional oscillation. The small
amplitude librations of a molecule within its crystalline potential
well are referred to as torsional oscillations. Molecular reorienta
tion will refer to a physical rotation of the molecule, e.g. a rota
tion of the NH^ molecule about its C^ axis through 2^/3 radians— from
one position of stable equilibrium to another.
The structure of liquid* NH^ has been determined by Xray diffrac
tion work performed by Kruh and Petz (39). The liquid x^as studied
under its vapor pressure at 199, 228, and 277 K. Radial distribution
functions were calculated for all temperatures. They interpret the
radial distribution function at 199 K to indicate that on the average
an ammonia molecule has eleven neighbors, seven at a mean distance of
3.56 S and four at 4.1 S. The liquid structure bears a close resem
blance to that of the solid, a manifestation no doubt of the ordering
effect due to the polar nature of the molecules. One may be led to
the interesting speculation of a highly ordered liquid state when
confronted with these results, a speculation not supported by nuclear
relaxation results. Reorientation in the liquid state is indeed quite
rapid, 1 yy second, and appears to be of an isotropic nature.
^•^' Introduction to Nuclear Relax ation vi.a Molecular Motion ■
Our objective in this brief introduction is to review for the read
er the various contributions to the measured spinlattice relaxation
58
T ,„EXP
rate, l/T^ . The application of detailed theory to the experimental
results for ammonia in order to separate the various contributions
will follow in later sections. We should emphasize that this dis
cussion applies both to the solid and the liquid state. The physical
^ difference between the molecular liquid and solid states is character
j Ized by the modes of molecular motion which are thermally activated.
I It is the determination of what modes of molecular motion are present
at a given temperature for which we strive. A much more detailed
account of the problem than that given here may be found in lectures
by Powles (40) and Bloom (41,42).
The problem at hand is quite formidable: To separate the various
contributions to the spin lattice relaxation time, T^^, and interpret
what mode of molecular motion leads to a specific contribution. The
1 separation is conveniently effected through the assumption that
Eq. (3.69) may be written in the form:
M (t) = M •
z o
f M  M(t=0)
1  —^ EXP(t/Tf^^)
o
(4.1)
where
EXP ? 1 • ■ (4.2)
1 ^ ^1
Proof of the validity of this assumption is not a trivial matter theo
retically, but empirical confirmation leads to very good results and
the assumption is widely used. It is obviously not applicable when
the recovery of M^(t) is not describable in terms of a single exponen
tial.
For diamagnetic m.aterials, the sum in Eq. (4.2) consists of the
following four terms :
59
. . .intra d ,
1 **'' '■''■1 ~ *^"^ intramolecular dipoledipole term: This term
arises from dipoledipole interactions among the spins in a single mol
ecule. It seems quite generally valid to assume that the molecular'
!
; bonds are rigid; therefore, magnetic fluctuations which stimulate Zee
^ man transitions are caused strictly by fluctuations in the angle e
I
(Eq. (2.6)). In solids this term is usually much larger than that due
to intermolecular dipoledipole interactions, but in liquids both terms
are typically of the same order of magnitude.
„ V ^ .inter d .
a. J x/i^  the intermolecular dipoledipole interaction
term: The magnetic field fluctuations in this case are produced by
dipoledipole interactions (Eq. (2.6)) betx^een spins on different mole
cules. If molecular selfdiffusion and reorientation are completely
independent, the effect of this term is governed only by the frequency
and magnitude of molecular collisions. This term will then contribute
information concerning molecular selfdiffusion.
SIT
C.) l/Tj^  The spin rotation interaction term: The spin rotation
interaction is the interaction of a nuclear spin magnetic moment with
the magnetic field produced by a rotating molecule. Fluctuations in
this magnetic field caused by collision modulated molecular rotational
states, i.e., changes in the molecular J quantum number, stimulate
nuclear Zeeman transitions if the fluctuations are rich in the Larmor
frequency Fourier component. This term is usually important only
around the critical point of the liquid. For the case of ammonia,
SIT
l/T^ has been measured by Smith and Powles (43) and found to be of
little importance in the normal liquid region. Thus we are justified
in not considering the spinrotation interaction in this work.
D.) 1/Ti  The electric quadrupole interaction term: If a nuclear
)
■ 60
spin has an electric quadrupole moment and its molecular environment
is such that a nonvanishing quadrupole coupling constant exists, then
molecular reorientation causes a reorientation of the field gradient
tensor (see Sec. 2.3.3.) which produces random field gradient fluctu
ations at the nuclear site. Under these circumstances the field gradi
ent fluctuations are usioally a much more effective relaxation mecha
nism than any other because the quadrupole interaction energy is, in
most cases, much larger than that of the dipoledipole interaction.
Since this relaxation process depends strictly on rigid molecular re
orientation, the information one obtains from IjT^ is the same as that
, ^ . J J 1 /„intra d
obtained from 1/T
The actual separation of the contributions mentioned above must
await the introduction of suitable physical models with which one may
mathematically represent the physical situation. We now proceed to
introduce such models for the most ordered state and then we shall
continue to the liquid state.
4.4. Interpretation of Relaxation and Continuous Wave
Data from Protons in Solid Ammonia
4.4.1. Interpretation of Proton Relaxation Data
We first consider the spinlattice relaxation time data illustrated
in Figs. 10, 11, 12 and 13 in the form of Log T vs. 10 /T, where T is
the absolute temperature of the sample. The experimental data points
are represented by bars which correspond to statistical intervals of
99% confidence. The method used to process the data is explained in
the next chapter.
For the case of solid NH„, Eq. (4.2) may be. written:
61
J L
0>
o
T1 — I — r
mill I I \ III I I I I I I 1 1 1 1 1 I I I t ,p
(O
— G
T I M i I I — r
Mil! I — I r
TTTTTTT — T
O
t=ss) 1
In
.^
i
to
"5
u
^.
S
<M
o
•■■
»
x:
»
1
O
'^
5
O
,t
•^
>.
■©
"o"
"S
a:
"—
(Ti
=3
1
<
s
00
LU
UJ
N
_l
O
<
U)
>
in
o
I
o
a
il
)
62
J ' ' 'MM I — 1 L ! " I I I I I I ■ I I I 1 I I I I I (O
_in
.■sf
c
o
a.
o
o
ro
_0J
i
N
X
or
<T>
T I I rmm — i — i r
—
00
—
e
— ~
N
©"
to
1 1 1 IT
1 ( r ,
1 1 1 1 1 1 1 1 1
^J
UJ
£r
I
<
cr
Q.
LU
I
LU
I
_J
O
0)
CD
<
>
I 39S) ,^.i
O
Q.
iZ
63
J L
m I I I I t I
III I I M I I.
11" I I I I
)
©_
e
©r
N
e
)
O
c
o
a.
a
o
o
a>
T— T
TTTTT 1 — i r
TT
"TT— 1 — r
J_U I 1 I I
o
(39S)Jl.
(0
lO
ro
CVJ
2 <
00
tr)
in
t^
IT
r>
<
UJ
OL
LJ
ai
_J
O
fS
CO
>
O
H
O
cr
il
64
J I I
Nil — l__J MM I I \ — I ! M I I I . t f
N
^
^
c
o
O
o
o
V
I
I
9
.U3
in
.^
fO
OJ
0»
co
h
co
in
1 — I — r
rrrm— I — i — i mm— r— r— i — r
"]]~n I I I — I — r'^
o
a:
UJ
Q.
■s
UJ
t
UJ
1
D
_l
O
CO
CO
<
>
l~
o
q:
a.
to
65
__L_ = L + 1 + 1 + 1 , (4, 3)
^EXP ^intra d ^intra d inter d inter d
1 Ip? Ipn Ipp ipn
where the letter subscripts refer to protonproton (pp) and proton
nitrogen (pn) interactions. This is a distinct assumption proved by
experimental results to be valid over most of the temperature region
studied, i.e., the recovery of M (t) was found to be describable by
Sq. (4.1) at very nearly all temperatures. Eq. (4.1) was not rigor
FXP
ously valid in the immediate vicinity of the T minimum; this is
explained later.
A model describing molecular motion must be chosen by appeal to
previous experiments. We must consider any process which causes mag
netic field fluctuations at the nuclear sites. In order for a process
to stim.ulate Zeeman transitions, it must have a strong Fourier com
ponent at the transition frequency of the nucleus; thus lattice vi
brations and molecular torsional oscillations can usually be neglected
as being of far too high a frequency to induce transitions. Reding and
Hornig (38) measure molecular torsional frequencies on the order of
13
10 Hz whereas the Zaeman transitions for our case occur at fre
quencies on the order of 10 Hz. On this basis, torsional oscillations
and lattice vibrations are neglected; the justification for this ap
proximation rests on the final results. Through comparing results
for other solids (see Abragam (7), pp. 45158) we are led to considera
tion of large scale molecular reorientation. As mentioned in Sec. 4.2.
each molecule is hydrogen bonded to its six nearest neighbors and the
bond energy of all bonds is presumably equal. A moment's considera
tion of the lattice structure (Fig. 9) will convince the reader that
the lowest energy m.olecular rotation transition should presumably
66
occur about the symnetry axis of the molecule. This presumption is
well supported by the ^^N quadrupole resonance work of Lehrer and
O'Konski (14). We thus consider the following model: The proton
nuclear relaxation is governed exclusively by inter and intramolecular
dipoledipole interactions modulated by the molecule undergoing hin
dered rotations about the symmetry axis to the three possible posi
tions of stable equilibrium. The reorientation process is assumed to
be a stationary random process which may be characterized by a correla
tion time T^, which is roughly the average time between molecular re
orientations. We assume that the process is thermally activated and
T^ obeys an equation of the form (see Abragam p. 455),
T^ = T^EXP{E^/RT}, (4.4)
where t„ is the value of x^ at T = <. , R is the universal gas con
stant, and E^ is the activation energy for the process, expressed in
calories/mole. Eq. (4. 4) is called the Arrhenius equation. For this
particular case, E^ is approximately the height of the barrier hinder
ing molecular reorientation.
As a first approximation to Eq. (4.3) we assume all terms on the
right hand side are negligible but the first. Hilt and Hubbard (HH)
(44), using density matrix formalism, have developed the theory for
nuclear magnetic relaxation of equilateral triangular configurations
of three spins with I = 1/2 undergoing hindered rotation about the
symmetry axis. They consider both the cases of a molecule undergoing
random jumps between three equilibrium positions and a m.olecule under
going stochastic rotational diffusion about its symmetry axis. The
results of the txro cases differ only in the definition of t . We
67
assume the first case, where
T ^
c 3v ' (4.5)
and V Is the probability per unit time for a transition between points
of stable equilibrium. The calculations involve autocorrelations of
) each dipoledipole interaction and cross correlations between differ
ent dipoledipole interactions. Cross correlation effects lead in
general to the necessity of describing the longitudinal relaxation
(spinlattice relaxation) in terms of a sum of decaying exponentials
rather than a single exponential (45). The cross correlation effects
are quite small in the case of the molecule undergoing isotropic re
orientation (46) , as will be shown later in this work in connection
with liquid NH3, but may be large for the case of hindered uniaxial
reorientation (47) .
With cross correlations taken into account, (HH) find that M^(t)
■ after a ./2 pulse is described by a sum of four decaying exponentials
with different time constants q./T' and different weighting factors
C., viz.
where
4
M^(t) =MJ1 ICEXP(q t/T')], (4.^)
J=l ^ J
T' 6, 4^2
^ = Vo/y« ' ■ (4.7)
and r^ is the distance between nuclei. Eq. (4.6) is dependent on the
angle of orientation 3 of the molecular symmetry axis with respect to
the Zeeman field, and both q.. and C. are tabulated in ref . (44) for
eleven different values of g. They also numerically average over the
angles, and provide graphs of the quantity Ln[(M (t)  M )/M ] vs
2 00
t/TV. These graphs are convenient for comparison of the theory with
68
experimental work on polycrystalline samples.
Fig. 14 depicts a comparison of this theory with relaxation data
for protons in solid NH^ at T = 110 K. The data fit the theoretical
values better if the sum of exponentials in Eq. (4.6) is multiplied by
a single exponential to take into account, in an ad hoc manner, the
effects of intermolecular dipoledipole interactions. One must how
ever assume a quite large intermolecular protonproton interaction to
obtain a reasonable fit. We are led to believe that observed non
linearity may be explainable by the (HH) theory because the effect
was strongest, as predicted by the theory, in the region of tempera
EXP
tures close to the T^ minimum. The equipment in this laboratory is
not suitable for a detailed study of this effect, but we are able
nonetheless to obtain an analytic expression for t vs . T which is
thought to be quite accurate. We will also compare our T^^ results
to the HiltHubbard formulation which neglects crosscorrelation
effects.
With no cross correlations taken into account (HH) find M (t) is
z
described, following a ^/2 pulse, by the following equation:
M (t) = M [1  EXP(t/T^"^^^ '^)1
z o Ipp ' ^ '
where
9 YS T^
^intra d 16 6
Ipp ^o
A 9 /
(1  cos 6) _j_ (1 + 6cos B + cos 6)
1 + (u T )'
o c
1 + (2aj T )
o c
(4.8)
(4.9)
Subsequently we refer to T^ values given by Eq. (4.9) as T^. To com
pare the values of T^ given by Eq. (4.9) with our experimental results
for polycrystalline NH.^, an average of the quantity (M  M (t))/M
o z o
must be made over all possible angles B, i.e.
<^W'^»*«>«ryttit[4.iUll» • ««^»U.,it •4>u.' .«Nai
69
;
TIME (msec)
2 4 6 8 10 12 14
16 18 20
2.2
24
2.6 •
Theory (co.i;) = 
Experiment 0,= 20.8 MHz
Temperature 1 10.2 K
Fig. 14. LNlMj RECOVERY) vs. TIME— PROTONS IN NH,
70
<R(t)> / ° ' '
\ M
o
1 ^
= 2 / EXP(t/T'^)sined3. (4.10)
o "^
This problem required the use of Simpson's numerical integration pro
cedure to evaluate the equivalent integral,
1
<l(t)>g = / EXP[t/T^]dx, (4.11)
where the substitution, x = cos3, was made. The integral was approxi
mated by 10 intervals, i.e., x = 0, 0.2, 0.3,.., 1.0, and the calcu
lation was performed using a HewlettPackard 9100A computer. This
integration procedure was carried out for at least three different
values of t for each value of o.^t^. For a given value of o^^x^, the
corresponding T^ was available from the slope of
Ln<R(t)>g = Ln / EXP[t/T^]dx (4.12)
. . o
vs. t. One may not rigorously define a single T,^ for the M (t) re
1 z
covery expressed by Eq. (4.11). We find however that Eq. (4.12) is a
quite linear function of t (see Fia 15^ anH a =>r,rr7o t*^ • 1
0. K. V.OCC rxg. xjj ana a single T governing the
relaxation is a good approximation.
Fig. 16 depicts Log tJ, calculated by the procedure above with '
r^ =1.651 S, vs. Log t^. The four frequencies used in the experiment
are represented. One should note that the T^ minima occur at o) t =. 5
'• o c
and that the curves corresponding to different oj^ are indistinguishable
in the short correlation time limit [(tu t )^ << ll .
o c ^ *
We now proceed to obtain an expression for t vs. T . From
c 1 ^
Eq. (4.9) we see that, in the long correlation time limit f((. t )^ » i)
c ^ o c' ' '
T^ is directly proportional to r^, but this is not a. rigorously valid
expression. Assumption: The actual physical situation is such that,
in the long correlation time limit, T^^ = kx^ . This assumption is ■
71
x/r
2 4 6 8 10 12 14 16 18 20 22 24 26 23 30
\i
\
\
\
 .2
%\ IV\_/vrt/^i
\\ l".i<J  U.UUI
A («.t)'= 100
\\
\\
 .4
\ \
\ \
\ \
\ \
"^ \ »
.6
\ \
\ \
\ \
\ \
\ \
.y^  8
\ \
•*Os, o
\ \
2
\ \
Sx
\ \
z
\ \
_J
\ \
1.0
\ \
\ \
\ \
^ \
\ \
\ \
1.2
\ \
^ \
\ \
\ \
\ \
1.4
\ \
\ \
\ \
\ \
.<
\ \
V \
1.6
\ \
\ \
\ \
^ \
\ \
1 A
\ ^
Fig. 15. LN<(R(t)) vs. TIME^T'
72
I ' ' ' I '
n ' I I I i L
■X,H,I I I ' '
o
in
. V
2
2
O
I
<
LU
IT
CC
o
o
<
h
z
UJ
o
Ll)
(r
2
O
o
(r
a.
b
<
I M I I I r
1 i I I I i — r — r
(09S) 'i
11 I I i~r
o
cr
o
LlI
X
CO
en
iZ
73
quite well justified experimentally in the region of interest as is
shown by the high linearity of the plot of Log T^^^ vs. 10^/T in the
low temperature limit. The reader should refer to Figs. 10, 11, 12
and 13 for support of this statement. Substitution of T^^ = kx into
1 c
Eq. (4.4) yields, after some obvious manipulation,
LnT  LnT + — ^ ~. 4,13)
10 R ^
Using standard linear regression techniques to fit the T^^ data
points in the linear low temperature region of Figs. 10, 11, 12 and 13
to Eq. (4.13), one obtains four values of E which should be the same
3.
within statistical error. A weighted average over the four values of
E^ is performed using as weighting factors the inverse of the product
of a 99% confidence factor with each estimated standard deviation of
E^. The following value of the activation energy is obtained:
E^ = 2.30 + 0.02 kcal/mole. (4.14)
We choose the low temperature limit for this determination for two
reasons. The foremost reason is that the (HH) exact theory predicts
very nearly single T^ recovery in the low temperature limit (see Figs.
3 and 4 of ref. 44), whereas in the short correlation time limit this
is not true. Thus we expect our experimentally determined T^"^ to be
more accurate in the long correlation time limit. Secondly, there is
additional risk, as the temperature is increased, of thermally acti
vating additional modes of molecular motion. In this case the relax
ation would be governed by additional correlation times and Eq. (4.13)
would no longer be valid.
To determine the constant t°° in Eq. (4.4), we appeal to the best
'"=<' ift" r^j ^ e™.i4lJr~.* itr**''
74
experimental data obtained (to^ = 2Tr x 20.8 MHz). Before continuing,
EX?
we must mentxon that Ti_ was obtained in each case by a linear re
gression fit of experimentally obtained values of Ln5.(t) vs. t (R(t)
as defined by Eq. (4.10)). Furthermore, values of LnR(t) used seldom
exceeded 2. The reader may easily convince himself that the minimum
T^ value, obtained from Figs. 4 and 5 of ref. (44) by fitting the best
straight line through the curves for Ln<R(t)) „ > 2, occurs at
p
"o'^c ^ "^' ^^S^' ^ ^^^ ^ represent exact results of the (HH) theory,
i.e., when crosscorrelation effects are considered. Although our
T,EXP ,
r^ values are meaningless as an exact recovery constant in the re
gion oj^T^  1, we nevertheless may rigorously determine the tempera
ture for which co^t^ = 1 from the minimum in the curve T^'^ vs. lO^^/T.
Inspection of Fig. 12 reveals that this minimum occurs at 10"^/T=9. 2+0.1.
1 3
Using T^ = 0)^ at 10 /T and Eq. (4.14) in Eq. (4.4) we obtain a value
T^ = 1.85 + 0.43 X 10 sec. The confidence limits placed on x
account for the maximum estimated error in determining both E (Eq. 4.14)
3.
3 EXP '^
and 10 /T at the minimum of the T^ vs. 10 /t curve. We thus obtain
the following analytic expression for t vs. T:
c
T^ = [1.85 + 0.43 X 10"^^]EXP[1155/T]. (4.15)
Fig. 17 depicts Log ^^ vs. T calculated from Eq. (4.15). The in
verse halfwidth betx^een points of inflection obtained from the CWNMR
data is plotted on the same graph to provide some perspective for the
reader as to the extremely rapid temperature dependence of Eq. (4.15),
One obtains an intuitive feeling from Fig. 17 for the sharpness of the
observed linewidth transitions attributed in the literature to ther
mally activated molecular rotation.
75
J ,
UJ
a:
z>
<
ai
Q.
S
UJ
I
(0
>
O
I
<
UJ
tr
o
z
g
<
I
O
a:
Q
UJ
cr
i
X
o"
o
((08S) ';, rsOl
76
We now return to the additional terms in Eq. (4.3) and attempt to
EXP
estimate the effect of each on T . The closest nucleus to a given
set of protons is the molecular N nucleus, which is positioned a
distance 1.005 A from each proton. Interactions of this nature con
tribute the 1/T, ^ term in Eq. (4.3). To theoretically solve this
problem exactly for our model would be quite involved and has not been
done. One would expect a set of coupled equations for the proton M
z
recovery such as those in Abragam (7), p. 295. Note that the strength
2 2 — fi
of the nitrogenproton interaction is proportional to y Y r . The
p n pn
second moment expressed in angular frequency has this sam.e form—pro
portional to the interaction energy. 1/T is also proportional to the
interaction energy; therefore, we expect the ratio of the respective
contributions to the total second moment to give a rough estimate of
the upper limit of the ratio of respective contributions to the relax
ation rate. From a second moment calculation (Sect. 4.4.3.) we find,
^ ,^intra d 2 intra
, /intra d " J\ntra = ^O^^' C^^^)
Ipp pp
The various contributions to the second moment were calculated using
the ReedHarris (33) distances in the corresponding terms of Eq. (3.37)i
The term 1/Tj_ , corresponding to Eq. (4.16), is over 100 times
smaller than Eq. (4.16) and is therefore ignored.
It must be noted, however, that the quadrupole relaxation mecha
nism may cause additional fluctuations of the N nucleus which are
not related to the hindered rotation relaxation mechanism. For ex
ample, molecular torsional oscillations about any molecular axis may
induce Zeeman relaxation of the nitrogen nucleus through the
)
11
quadrupole coupling aiechanlsm, thereby producing fluctuations at the
proton sites which are totally uncorrelated ^jith hindered rotation.
14
We attempted unsuccessfully to find the N pure quadrupole resonance
in solid NH with the pulse apparatus in order to examine this possi
bility.
Finallv we concern ourselves with the 1/T, term. To this
Ipp
writer's knowledge, no detailed theoretical prediction of this term
has been made for any solid. This would be a quite complex theoreti
cal problem and we are content to present only an upper bound esti
mate of the effect of this term relative to the intramolecular pro
tonproton interaction. The simple ratioofsecondmoments argument
used previously is probably a reasonable estimate in this case be
cause, at any point during a hindered rotation, the average intermolec
ular protonproton interaction strength remains roughly constant.
Using this argument, we find
 ,_inter d „2 inter
1/T M
^ 7  ^7— = 0.24. (4.17)
^^^mtra d ^^^2 xntra
Ipp " pp
We are able to estimate from Eqs . (4.16) and (4.17) that the maximum
additional contribution to the total proton relaxation rate from proc
esses other than intramolecular proton dipoledipole interactions is
approxim.atelv 30%.
»
EXP
With this inform.ation in mind, we reexaud.ne the Tf data pre
j.
sented in Figs. 10, 11, 12 and 13. The solid black points among
experimental points were theoretically calculated by first using
Eq. (4.12), and the approximation outlined, to find T^ for a given
value of (J T = x, then substitutins; t = (oj ) x into En. (4.15) to
o c CO . \ /
78
3
find 10 /T. The intramolecular protonproton distance used is that
J "^ °^ ^^f (33) » ^o = 1.651 £. The agreement is quite good in general
in both the long and short correlation time limits — recall that the
T,EXP
T^ poxnts have truly little physical meaning in the region about
J '^o'^c ~ ^ ^s ^ result of the high nonexponentiality present in the
recovery. We nevertheless plot them to illustrate the fairly sharp
minimum from which xj" was determined. The reader should refer to
j Fig. 12 for illustration of the points discussed since the data taken
at 20.8 ^ffiz were considerably better than at other frequencies. Note
EXP
that the minimum in T^ does not occur at the same point as the
. . . c
minimum m T^. The reason for this was explained previously.
As mentioned previously there is a discrepancy in the literature
regarding the solid state intramolecular protonproton separation in
the ammonia molecule. The Xray work by OlovssonTempleton (12) re
, veals a value of r^ = 1.817 S whereas the neutron diffraction work by
ReedHarris (33) on ND gives the value r = 1.651 1. We wish to
J o
offer rather unique support for the (RH) value through our T^^
measurements. One may observe from Fig. 4 of the HiltHubbard (44)
paper that a single exponential recovery is a very good approximation
I to the exact theory when ((j^t^)^3< 100. We observe from this figure
that when (oj^t^) = 100, T^  13.2 T', where T^ is the effective
intramolecular spinlattice relaxation time and T' is defined by
J Eq. (4.7). For oj^ = 2^ x 20. 8 MHz we calculate the following values
for the intramolecular contribution to T : a.) Using the (0T)
value for r^  T^^^ = 109.0 msec, b.) Using the (RH) value for
'iRH ~ ^^'^ msec. Substituting x = lO/o) into Eq. (4.15), we
r  T
O x!\n. ■ ° C """' "o
calculate lO^/T = 11.2 + 0.2. These calculated values appear as
79
horizontal error bars on Fig. 12. Note the large separation of the
two values. The T^ measurements permit the same high resolution as
second moment measurements due to the same spatial dependence; i.e.,
the sixth power of the nuclear separation. We have estimated the
J total contribution to 1/T^ from the sum of Eqs. (4.16) and (4.17) to
I be roughly 0.30/T^^^^^ ^ Using the value T^^ = 61.4 msec, calcu
lated previously, one finds the theoretical prediction for this con
tribution to range from 0.04/1^^^ to 0.60/tJ^ with a most probable
I ^^^"® ^•■'■^''^iPvA Li'^^wise using the value T^^^ = 109 msec, one finds
that relaxation mechanisms other than intramolecular protonproton
j must contribute from 0.85/1^^^ to 186/1^^^ with a most probable value
of 1.27/Tj^Q^. We see that the (RH) value of r yields results com
patible with both theoretical predictions and our experimental T^^
results, whereas the (0T) values do not.
The discrepancy in r^ found by the two methods is quite possibly
not due to experimental uncertainty, but rather a manifestation of
the different physical quantity measured by the two methods. As men
tioned previously, Xray diffraction measurements give information
concerning density peaks in the electron cloud, not the nuclear posi
tion as does neutron diffraction. Recall that molecular bonding in
NH^ is such that the molecular electron cloud is heavily concentrated
about the nitrogen nucleus, leaving the protons relatively bare. In
^ the solid each proton forms a hydrogen bond with a nearest neighbor
molecule through mutual electrostatic attraction between the proton
and the lone electron pair of the neighboring molecules. It seems
quite feasible that the difference synthesis, performed by Olovssen
Templeton, revealed the electron cloud surrounding each proton but
.;
)
80 .
weighted by the distorted free electron pair distribution of the hydro
gen bonded neighbor. This would explain the apparent lengthening of
the NH bond.
4.4.2, Interpretation of Proton Li ne width Data
We presented in Sec. 3.4.1. a qualitative picture of the phenom
enon of linewidth reduction in solids through molecular motion. The
concept of an average lifetime x of a nuclear precessional state was
^ CO
introduced and we stated that motional narrowing begins to occur when
T  T^. (4.18)
We also showed that T^ is roughly equal to the inverse of the linewidth
6u. The condition for the onset of a motional narrowing transition may
be written,
T  (oo))""'. (4.19)
OJ
In an absolutely rigid solid, x is governed by spontaneous Zeeman
OJ
20
transitions, viz. x ~' 10 seconds. Molecular motion stimulates
OJ
transitions quite effectively, thus reducing x to the average time
CQ
between molecular m.otional transitions, x^hich we have labeled x^.
With these thoughts in mind, consider Figs. 17 and 18. Fig, 18
depicts the halfwidth of the CTJl^'IR line between points of inflection
vs. the absolute temperature of the sample. Fig. 17 depicts our
calculated values of the correlation time for hindered molecular m.o
tion about the syminetry axis vs. absolute temperature (heavy line),
and (6a))~ vs. absolute temperature (light line). The intersection
of the two curves in Fig. 17 gives one the absolute temperature for
which the linexridth. should begin to undergo a sharp reduction. The
value obtained in this mianner (''59K) agrees quite favorably with the
onset of the first linewidth transition as depicted in Fig. 18. We
81
J
J
^ (£) ^ (NJ O CO 03
(ZH>i) HiaiM3NI"l NQlOyd
tr.
<
UJ
a.
UJ
>
9
2
Z
o
H
O
q::
Q.
00
c:
82 .
conclude on this basis that the linewidth transition which is centered
at 65 K is produced by hindered rotation of the NH molecules about
their synnnetry axis. A more accurate statement of the origin of this
linewidth transition will be made in Sec. 4.4.3.
J One observes from Fig. 18 two additional points of interest dis
cussed In Sec. 3.4.1.: a.) The broad, reasonably flat plateau, and
b.) The occurrence of an additional linewidth transition, beginning
at T = 170 K, but interrupted before completion by the solidliquid
phase transition. The second linewidth transition is of particular
i
interest because it marks the activation of a motional process com
pletely distinct from symmetry axis hindered rotation. We speculate
that this new process is either isotropic reorientation of the mole
cules, or molecular selfdiffusion— note that molecular selfdiffusion
must involve isotropic reorientation. It is not possible to deduce
precisely what process occurs in this region from the measurements
1 made; however, one may obtain a very good estimate of the correlation
time involved from Fig. 17 and Eq. (4.19). By direct analogy with
the previous case, we consider T  180 K to mark the temperature at
which Eq. (4.19) is valid, i.e., a plot of i ^^, the correlation time
for the second process, vs. absolute temperature will cross the (6(j)"'
curve of Fig. 17 at T  180 K. This point on the graph corresponds to
^c2 "^ '^ ^ 10 sec at T = 180 K. (4.20)
It is also possible to determine the activation energy for this
second process to an accuracy of approximately 10%. A number of quite
involved theories have been developed (9,48) to relate the linewidth
and second moment change at a transition to the correlation frequency
J
83
of the process involved. The theories are difficult to use because
many very precise measurements are required. Waugh and Fedin (49)
make suitable approximations to a very simple theory and arrive at
the following approximate relationship relating the center temperature
J of a linewidth transition to the activation energy of the process
involved :
^a = ^7 T^' (4.21)
where E^ is expressed in calories/mole and T^ is the absolute tempera
ture of the transition midpoint. This relationship agrees within 10%
of more detailed (but not necessarily more accurate) theories. For
comparison, Eq. (4.21) gives a value of 2.4 kcal/mole for the activa
tion energy of molecular hindered rotation about the symmetry axis;
a 5% deviation from our previous calculation (Eq. (4.14)). By use of
Eq. (4.21) we estimate the activation energy of the process causing
the second transition to be
E^2 ~ '^•2 + 0.7 kcal/mole. (4.22)
The question of why the second motional process did not appear in
the T^ measurements has probably occurred to the reader. Ammonia
provides a nice case for discussion of the relative sensitivity of the
two NMR techniques, which we used, to various frequency regions of
molecular motion. We take this opportunity in answer to the question
J ■ raised to briefly discuss this point. Comparison of Figs. 16, 17 and
18 provides one with the key to this question. The linewidth measure
ments of Fig. 18 are observed to be extremely sensitive to molecular
motion onlx in the region where Eq.. (4.19) is approximately satisfied
(refer to Sec. 3.4.1.) and are quite insensitive to molecular motion
J
84
of other frequency. The linewidth is a characteristic of the solid
itself, so the frequency region of molecular motion that we may ob
serve through C\mifm. techniques is fixed by the characteristics of the
material. T measurements are much less limited in this respect. We
note from Fig. 16 that T is a highly sensitive function of to x , and
1 o c
the nuclei couple most effectively to molecular motion when the follow
ing relationship holds:
■"c " ^%^'^ (^23)
Eq. (4.23) is to TNMR what Eq. (4.19) is to CIINMR, but there is one
very important difference: The value of to is determined uniquely by
the nuclear gyromagnetic ratio and the externally applied magnetic
field. Thus the region of x we may study by Tms. is a laboratory
problem, limited only by how cleverly we design our electronic appa
ratus. The literature abounds with clever tricks to study values of
T^ not usually accessible. A book describing these techniques in
detail is badly needed, but this is not our purpose and we must trun
cate a very interesting subject, only barely introduced.
4.4.3. Calculation and Interpretation of the Proton Second Moment
We used the results of a second moment calculation when interpret
EXP
mg the T^ data; the calculation is now presented. The ReedHarris
(33) lattice and molecular parameters are used. Since all protons and
N nuclex are equivalent, we use Eq. (3.37) and consider: Intramo
lecular contributions and contributions from as far away as third near
est neighbor protons. The geometrical configuration is shown in Fig.
9, and the various contributions calculated are given in Table I.
85
J
J
Table I. The Independent Contributions .to the Proton Second
Moment for Solid NH^
Contribution of:
^^{(P)
Intramolecular protons
35.36
Molecular N
2.15
Nearest neighbor protons (Molecule to which H bonded)
4.34
Third neighbor proton (Molecule to which H bonded)
1.58
Second neighbor protons (Protons mutually H bonded to
same N)
2.64
Total :
46.07 G^
The fourth nearest neighbor protons are nearly 1 A farther away
than the third nearest neighbors. We estimate the total neglected
2
intermolecular contribution to be less than 2 G . The second moment
contribution from the intramolecular nitrogen was calculated assuming
that the predominant contribution to the nitrogen Hamiltonian is from
the Zeeman effect. This is of course not true in the case of NH ■ the
nitrogen quadrupole interaction in solid NH„ is roughly equal, at the
magnetic field available, to the Zeeman interaction. That this may
lead to error in the calculated second moment is brought out in ref.
(50). We ignore this effect on the basis that it will probably con
tribute nothing additional to our final interpretation of the data.
The experimentally measured second moment was calculated from
CWN14R data for a number of different temperatures throughout the solid
range and was, in each case, corrected for modulation broadening.
The experimental points are illustrated as solid vertical lines in
Fig. 19. The points are unfortunately sparse, but we will nevertheless
86
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87
be able to make a number of important observations. Two additional
points (circled) were obtained by Gutowsky and Pake (51) and Rabideau
and Waldstein (52) at temperatures of respectively 90 and 75 K.
Clearly the most startling contrast with which one is faced is that
4 between the slowly varying second moment vs. temperature, illustrated
in Fig. 19, and the abrupt changes in the linewidth vs. temperature,
illustrated in Fig. 18. Secondly, and just as profound, is the large
difference in the predicted rigid lattice value of the second moment
(Table I) and that measured at temperatures as low as the ^He lambda
point. This large discrepancy is disturbing because our T measure
ments extrapolated to this temperature indicate a T on the order of
25
10 sec, which is longer even than the spontaneous transition prob
ability would indicate. Clearly the T^ measurements predict no hin
dered rotation at this temperature contrary to the second moment pre
diction that the molecule is rotating at an angular rate greater than
/• a % 5 —1
{hbO = 1.8 X 10 sec. . The resolution of this problem requires
use of concepts not yet fully understood— the concept of quantum me
chanical tunneling of the molecule through its hindering potential
barrier. Considerable effort has been expended at the University of
Nottingham on problems of this nature. The classic paper, referred to
many times in the literature, is that of Andrew and Bersohn (53) which
treats the problem of the effect of rapid classical rotation of a
J triangular configuration of nuclei on the apparent second moment of
the configuration. They find that the observable second moment of a
triangular configuration of spin 1/2 nuclei rotating about the C axis,
at a rate much faster than the square root of the second moment, ex
pressed in frequency units , is one quarter that of the. calculated rigid
88
lattice value. Their results seem to work well even in the tempera
ture region where classical rotation is impossible, if one assumes
that molecules in the torsional ground state tunnel through the hin
dering barrier at a rapid enough rate. This rather ad hoc approach
J has recently been placed on firm ground by Allen (54) who treats the
molecular motion quantum mechanically and finds that indeed for bar
riers of energy less than 3 kcal/mole (assumed temperature indepen
dent) the classical expression of Andrew and Bersohn should indeed be
' accurate, even at very low temperatures. Our previously determined
value of activation energy, E = 2.30+002 kcal/mole, places am
monia in this category. Allen and Cowking (55) have measured second
moments and T in some methylbenzenes and obtain results that appear
much like our Fig. 19, i.e., the long relatively flat plateau from the
melting point down, with the gentle change of curvature beginning at
approximately 70 K. One would expect our results to be similar to
theirs because first, the barriers hindering rotation in the methyl
benzenes are very close to that of NH„; and secondly, the inter and
intramolecular contributions to the total second moment are roughly
the same.
We now proceed to explain in detail our experimental second moment
values. The RabideauWaldstein point at 75 K is within experimental
2
uncertainty of our value at 75 K (they quote 16 + 1 G at 75 K.) The
J 2
^ GutowskyPake value of 9.7 G at 90 K is too low. This low a value
\ would require nearly complete averaging of the intermolecular con
tribution, a quite unreasonable requirement in view of the type mo
lecular motion occurring. We speculate that the state of the art in
1950 was such that they were not able to detect the reduced intensity
89
in the wings, thus substantially reducing their value for M2. Our
90:1 slgnaltonoise ratio at 90 K allowed good resolution of the
wings. The contribution to the total second moment from intramolecu
lar protonproton Interaction is assumed on the basis of ref. (54)
2 2
to remain at (35. 36)/ 4 G = 8.85 G . The contribution from the intra
molecular nitrogen may be calculated using the reduction factor
1 2 2
T(l  3cos y) [see Abragam (7), p. 454], where y is the angle between
the axis of reorientation (C_) and NH intranuclear vector. Using the
ReedHarris molecular parameters, we find y  71.53° and the reduced
2
value of this contribution to be 0.262 G . Thus the total constant
2
intracontribution is 9.1 G .
We calculated the theoretical second moment contributions using
the ReedHarris lattice parameters, measured in solid M)„ at 77 K;
2
and we assume these values to be correct for NH„. A value of 6.1 G
is found experimentally. Thus a very reasonable 2940% motional
averaging is required for the experimental and theoretical results to
agree. Since the condition for maximum motional averaging has long
since been met, the averaging of the intermolecular contribution
should remain constant. This being the case, we speculate that the
observed decrease in M„ vs. temperature, as the temperature is in
creased from 77 K, is attributable to linear expansion of the lattice.
We assume the intramolecular contribution is a constant. The linear
expansion coefficient was determined in the (0T) Xray analysis of
NH„ to be 11.2 + 0.5 X 10 K . Since the intercontribution to the
second moment varies as the sixth pox^er of the internuclear distances,
we expect that although this is a small effect it may be observable.
Assuming that all lattice distances expand according to the prescrip
90
tlon, r = r (1 + aAT) , where a is the expansion coefficient and AT is
the change in temperature, one finds M„ = M / (1 + aAT) . This rela
tionship is actually effectively linear in aAT because over the range
of validity, aAT is so small that terms of order higher than first
j are negligible. Thus, one has M„ = M^/ (1 + 6aAT) . We used the ex
perimental value of M at 77 K as M„ and the (0T) value for a. The
thick dotted line in Fig. 19 represents the results of this calculation.
I The doxraward trend of the experimental points beginning at roughly
165 K marks the excitation of an additional narrowing process. As
observed in connection with the linewidth data, this process is a much
higher energy process than that of hindered C„ rotation. The upward
trend below 60 K marks the region in which the classical hindered ro
tation becomes energetically impossible and the quantum mechanical
tunneling process becomes important. It appears that thermal expan
sion of the lattice provides a plausible explanation of the reduction
in M„ between 60 and 160 K.
We must now briefly consider T in the region below 60 K. This
region was not investigated in the course of this work because the
author was under the mistaken impression that tunneling phenomena have
little observable effect on T . Clough (56) pointed out that coher
ent tunneling will not effect T , and work by Eades , et al (57,58)
on methylpentanes indicated experimentally that very long T^ values
/ were consistent with motionally narrowed lines. The author was un
fortunately not ax<rare of the very interesting work of Allen and Cow
king (55) which demonstrates multiple T minima at low temperatures.
Until very recently, no satisfactory explanation of these minima exist
ed. This deficiency is apparently resolved in a very recent publica
91
tion by Allen and Clough (59) in which they derive an expression for
T^ vs. Temperature for the case of tunneling methyl groups. It is not
unlikely that the same phenomenom will be observed in NH^ and further
: 3
' work is planned in the low temperature region of ITO .
J The contrast between the M and linewldth curves was previously
mentioned. The high temperature transition is easily explained in
terms of activation of an additional motional process. This transi
tion is in fact reflected by the M^ data. One cannot brush off the
transition centered at 65 K quite so easily because it does not appear
to be reflected in the M^ data. One might wonder why a linewidth
transition occurs at all, in light of our M evidence that C rotation
occurs at a rate high enough to minimize intramolecular contributions
at temperatures as low as /^ 1 K. The answer is, no doubt, a manifesta
tion of a fundamental difference in the quantum mechanical tunneling
process vs. that of a thermally activated classical hindered rotation.
Since the linewidth is. a strong function of this difference, perhaps
a detailed study of linewidth in the transition region (region where
the rotational process is predominantly quantum mechanical to that
where it is predominantly classical) would offer some valuable infor
tion concerning the difference in the character of the motion. We
speculate that the linewidth transition occurs because the motion in
volved changes from a predominantly coherent character, at very low
J
^ temperatures, to a predominantly random character at higher tempera
tures.
92
4.5. Interpretation of Nuclear Magnetic Relaxation Data
from ^^N and Protons in Liquid Ammonia
4.5.1. Calculation of Molecular Reorientation Correlation Time
from T . Measurements on the I'^N Nucleus
The liquid state of any substance is characterized by an increase
J in the number of degrees of freedom and relative magnitude of molec
xilar motion. Although Xray data seem to point to a 'quasilattice'
type liquid structure for ammonia, we present evidence that indicates
a typical isotropic molecular reorientational time scale of approxi
mately 1 picosecond. This is not to imply that a 'quasilattice'
description, in which one assumes the liquid state to be such that a
molecule is held relatively securely for a short time before 'jumping'
rapidly to a vacent lattice position, is invalid for the case of NH„.
Quite to the contrary, a theory of this type should be well suited to
describe a strongly bonded polar liquid such as ammonia. Our results
do show, hoxjever, that the time scale for short range order in liquid
ammonia has an upper bound of roughly 1 picosecond.
Our objective at this point is to calculate values for the molec
ular reorientation correlation time. The fact that, in the NH mole
14
cule, there exists a relatively large N quadrupole coupling constant
14
(Sec. 2.3.3.) allows one to neglect all contributions to the N
Zeeman relaxation time (Eq. (4.2)) but the quadrupole term. This
technique for separating the molecular rotational correlation times
gives very good results and is discussed in detail by Moniz and
Gutowsky (60) and Abragam (7) . One assumes that the nucleus reorients
at a rate much faster than the inverse of the molecular quadrupole
93
coupling constant and thereby averages the quadrupole contribution to
the molecular energy to zero. This averaging process occurs because
the molecular reorientation is so rapid on the time scale set by the
quadrupole coupling constant that the nucleus cannot follow the field
gradient. When an external magnetic field H is applied, one has the
usual Zeeman spectrum with very rapid transitions occurring among the
states due to the quadrupole coupling.
The Larmor precession frequency in our magnetic field (H = 8.9 kG)
is oj = 2Tr X 2.74 MHz. Clearly from our work in solid NH^, our meas
urements are made in the short correlation time limit, i.e., tu x << 1.
o c
14
For the N nucleus (I = 1) , perturbed by an isotropic rapidly re
orienting sjTnmetric field gradient tensor, we find the following equa
tion for 1/T^ in Abragam (7), p. 314:
ft
(4.24)
We follow the typical practice of assuming an Arrhenius equation
for T (Eq. (4.4)). Eq. (4.24) may then be written
e qQ
^
T EXP[E^/RT]
y a
(4.25)
J
Taking the natural log of Eq. (4.25), one has an equation of the form,
LnT^ = e +
io\
10
(4.26)
which is fit, using linear regression techniques, to the experimental
EXP 3
points of T^ vs. 10 /T. The following analytic expression is
KXP
obtained for T as a function of temperature;
Ln(Tj ^^^) = 0.747  0.802p
(4.27)
The actual experimental points are shown in Fig. 20 plotted as
J
30
20
10
9
8
^ 7
«
^ 6
ap.
94
This work
Ref. (19)
4.2 43 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1
IQ?^T (K")
Fig.20. LOGjT*) vs. 10'/ (TEMPERATURE)
95
Log T^ vs. 10 /T. The solid line represents the linear regression
fit given in Eq. (4.27).
In order to use Eq. (4.24) we made the assumption that the molec
ular motion is isotropic. This assumption is justified on the basis
y of corroboration from previous work (19) from which the N relaxation
times vs. temperature were indirectly calculated. In ref. (19) the
NH, high resolution proton (see Fig. 8) lineshape was fit to the theory
of Pople (30). From the lineshape fitting procedure, both T^ of the
14
N nucleus and the indirect spinspin coupling constant J were deter
mined. The work reported in ref. (19) extends from 201 to 307 K. The
dotted line of Fig. 20 represents the results of their work over the
normal liquid region. The correlation between our results and those
of ref. (19) is excellent when one considers how very different the
two methods are. With this support of our procedure, we proceed to
'■i calculate first the reorientation activation energy, then x vs. T.
The reorientation activation energy is calculated from the linear
regression value of the slope of Eq. (4.26), i.e., from comparison of
Eqs. (4.26) and (4.27) one obtains Eq. (4.28). The indicated devia
tion represents the statistical 99% confidence interval.
e3 = 1.60 + 0.31 kcal/mole (4.28)
c
The values of T£ obtained from Eq. (4.27) are used in Eq. (4.24) to
calculate t vs. T. We perform this calculation for: a.) The free
2
molecule value (e qQ/fi) = 2it x 4.08 I'lHz and b.) The quadrupole cou
pling constant for the NH„ molecule in solid NH„, corrected for zero
2
point molecular motion (14), (e qQ/fi) = lir x 3.47 MHz. The latter
value is expected to be the' more correct, but again we let comparison
96
with previous work be the judge. The curves, t vs. T, obtained in
this manner are illustrated in Fig. 21. The regions of uncertainty
are calciiLatad using a rather pessimistic + 10% uncertainty in the T
1
data.
J Smith and Pox^7les (43) have investigated the proton relaxation
times in pure liquid NH„ and in mixtures of pure NH with ND . By
measuring the proton T in NH as a function of the ND_ concentration,
they were able to separate the intermolecular and intramolecular con
tributions to the experimental T . They assume that the correlation
time governing the intermolecular contribution is the same as the re
orientational correlation time x which governs the intramolecular
contribution. Finally, an expression is obtained for t, in terms of
a
the total dipolar contribution. This expression is corrected by
13
Powles and Rhodes (61) and they calculate a value of t, = 6.6 x 10
d
sec. at 202 K. They assume this correlation time to be identical to
T . The value of x they calculate lies less than 5% (well within our
experimental uncertainty) above our corresponding value of x , which
was calculated from Eqs. (4.27) and (4.24) using the quasistatic solid
state value of the quadrupole coupling constant (3.47 MHz.)
We interpret their corroboration of our data as support for the
existence of strong hydrogen bonding in liquid ammonia. If the liquid
were not strongly bonded, one would expect the appropriate quadrupole
coupling constant to be nearer the free molecule value than that of
the solid. It is interesting that they interpret this same data point
as support for the existence of little hydrogen bonding in the liquid.
Their interpretation is based on 1^ measurements made on deuterons in
heavy ammonia. This work was published by Powles, Rhodes, and Strange
;
97
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98
(62).. From the T^ measurement and the value of t stated previously,
they calculate the deuteron quadrupole coupling constant from an equa
tion of the same form as Eq. (4.24). The value they obtain for the
2
deuteron e qQ/h in ND (245 + 25 kHz) is close to that of a recent
) measurement for the NH D free molecule (63), 282 + 12 kHz; but higher
than an earlier measurement (64) for the ND free molecule (200 + 20
kHz). From comparison of their value with that of the NH„D free mol
ecule, the conclusion is drawn that little hydrogen bonding is present
in the liquid.
It is this author's belief that, although the PowlesRhodes inter
pretation of little hydrogen bonding in liquid ammonia is completely
J consistent with their observations, the more detailed analysis made
possible by their work, the work of Lehrer and O'Konski (14), and our
work strongly supports the Xray diffraction (39) interpretation of
strong hydrogen bonding in the liquid. Lehrer and O'Konski suggest
14
that the solid state shift of the N quadrupole coupling constant is
due primarily to distortion of the nitrogen lone electron pair. The
highly exposed position of the electron lone pair leads one to believe
that it should be very susceptible to distortion through hydrogen
bonding, thus providing a sensitive environmental probe. In light of
the 33% ionic character of the NH covalent bond and its short length
relative to intermolecular hydrogen bond lengths, one is led to the
J
speculation that by far the predominant contribution to the electric
field gradient at a deuterium (hydrogen) site is due to the electronic
structure of the nitrogen to which it is covalent bonded. Since the
i intramolecular ND distance remains nearly constant in the hydrogen
bonded solid, it seems likely that hydrogen bonding may affect the
99
o 1 /
e qQ/^. only through distortion of the molecular N electronic struc
ture. Furthermore, the effect is quite possibly less than the 15%
1 A 9
reduction of the N e qQ/!!.
Rabideau and Waldstein (52) have measured the quadrupole coupling
constant in ND„ from the quadrupole splitting of the Zeeman spectrum;
they find values of 156 + 7 kHz at 75 K and 72.5 + 0.9 kHz at 159 and
185 K. The reduction at the higher temperatures is attributed to
molecular reorientation of the molecule about the C„ axis, but they
assume that this motion is unlikely at 75 K. Our measurements on NH„
indicate a correlation time for C_ hindered rotation of approximately
7
9 X 10 sec. One would not expect this correlation time for ND„ to be
much different. Symmetry axis rotation alone may thus lead to aver
aging of the 200300 kHz quadrupole coupling constant. Furthermore,
molecular torsional oscillations will contribute additional averaging,
2
thus reducing the apparent e qQ/h to an even lower value. Until the
2
quasistatic solid state value of e qQ/h is determined for ND , an
absolute statement of the effects of hydrogen bonding cannot be made.
Powles and Rhodes also recognize this fact and do not support their
conclusion with this value.
From the information presented above, we are led to the following
conclusions as a possible solution to the problem:
a.) Liquid ammonia is hydrogen bonded as previously indicated
by Xray ar heat of vaporization work.
b.) The deuteron quadrupole coupling constant is not highly
affected when hydrogen bonding occurs.
;
100
4.5.2.. Interpretation of Proton T and T ^ Data from Liquid NH 
As was mentioned in Sec. 4.5.1., Smith and Powles (43) have meas
ured the proton T in pure liquid NH^ and in mixtures of NH and heavy
ammonia. Proton T^ has also been measured by Blicharski, et al (65),
as a function of temperature. Both of the experiments were performed
with the sample under its own vapor pressure. We repeated this work
with a constant absolute pressure of approximately 20 p.s.i. over the
sample. These measurements were made primarily to investigate the
paramagnetic impurity (mainly oxygen) content of our sample. T
measurements on protons in molecular liquids are in many cases a more
sensitive indication of oxygen contamination than mass spectrometry
(66). Our goal was to obtain results consistent with those of the
SmithPowles experiment.
It was noticed in the course of making the T^ measurements that
while T^ was on the order of five seconds, T^ appeared to be approx
imately three orders of magnitude smaller. At the time, this was
assumed to arise from magnet inhomogeneity. In order to be certain
that magnetic field inhomogeneity was the cause for the unusually
large difference in the two time constants (recall that in the case of
liquids, one usually finds T = T„) , we used the CarrPurcell (67)
technique to eliminate the effect of magnetic field inhomogeneity on
!„. Our T„ measurements by this method are crude but certainly in
error by no more than 20%. We find that indeed T„ << T , and the
explanation of the observed temperature dependence of T„ leads to a
quite reasonable but rough value for the protonproton chemical ex
change activation energy. We now proceed to consider the effect on
T^ and T„ of each interaction listed in the summary at the end of
101
Chap. II.
EXP
One may write a rate equation for T^ which is completely anal
EXP
ogous to that of Eq. (4.2) for T , i.e.,
T. k T.
X 1
where the i denotes either T or T and the terms present in the sum
are defined as follows :
„1 _
dipoledipole interaction.
T = the contribution from the intramolecular protonproton
2
T. ~ the contribution from the intramolecular protonnitrogen
dipoledipole interaction.
3
T. = the contribution from the intermolecular protonproton
interaction.
4
T. = the contribution from the indirect spinspin interaction.
Intermolecular NH dipoledipole interactions and contributions
through an anisotropic chemical shift tensor are neglected. The jus
2
tification for neglecting the former will be seen when the T, term is
considered. We have no information concerning the existence of an
anisotropic chemical shift in NH„, but a calculation assuming a quite
large anisotropy produced a negligible contribution to T..
The intramolecular protonproton dipoledipole interaction term
T^ has been calculated for the case appropriate to NH (three spin
1/2 nuclei arranged one on each corner of an equilateral triangle) by
Hubbard (46) using density matrix techniques. He assumes that the
molecule undergoes isotropic stochastic reorientation, characterized
by the correlation time t , and obtains the following expression for
the recovery of the nuclear magnetization along H :
102 .
R(t) = ^— 3J ~ = 9.192 X lO" EXP[t/T ] + 9.908 x 10"'EXP[t/T ],
o ^
where, —= 4.001 x 10""'° t ' (4.30)
b
and 7^ = 9.585 x 10"'° t .
i c
c
The numerical values were calculated using free molecule geometrical
parameters. Note that although the recovery is governed by two ex
ponentials, one approximately twice the other; it is a very good
approximation to simply drop the first term on the right hand side of
Eq. (4.30) because its weighting factor is two orders of magnitude
smaller than that of the second term. In the same spirit of approxi
mation we approximate the weighting factor of the second term as 1,
and the resulting recovery equation becomes:
R(t) = EXP[9.585 X IQ'^x^ t]. (4.31)
We thus have the following expression for T :
■\ = 9.585 X 10'° T . (4.32)
^1
Richards (68) has proved that My* = R(t) for the same physical approx
imations used by Hubbard (46). Using these results, we have
y = ^ = 9.585 X 10'° T . . . (4.33)
1 2
Abragam (7) develops the theory necessary to estimate, to a good
approximation, the effect of the intramolecular nitrogenproton
dipoledipole interaction. The two nuclei are assumed rigidly fixed
in a diatomic molecule, which undergoes random isotropic reorientation.
Our situation is of course different from this one, but we presume
that the relatively strong intramolecular protonproton coupling would,
103
if anything, tend to reduce the effect of any nitrogenproton coupling.
We speculate that what we calculate here will closely approximate the
2 2
upper limit of T^ and T„. Abragam obtains a set of coupled differen
tial equations for the recovery of the respective nuclei. The equa
tion governing the proton M recovery is,
rr = 6.03 X lO^^T (M (t)  M ) + 8.5 x 10 t (S  S ), (4.34)
at c z o c z o
where S (t) describes the nuclear magnetization recovery of the nitro
gen nucleus. The constants were calculated using free molecule geo
14
metrical parameters. In Sec. 4.5.1. we presented the N relaxation
tim.e data. On the proton relaxation time scale, ^ 10 seconds, the
quantity (S (t)  S ) may be neglected because it approaches zero, via
the electric quadrupole interaction, approximately three orders of
magnitude faster than the first term. Thus, in this approxim.ation,
~ = 6.03 x 10^ T . (4.35)
T^ '^
The transverse relaxation time is calculated from Eq. (89), p. 296 of
Abragam. It is
— = 2.83 X 10^ T . (4.36)
T ^
^2
To calculate the intermolecular protonproton contributions, we
appeal to additional work by Hubbard (69) . In this paper Hubbard
3 3
calculates T^ and T„ assuming: a.) The short correlation time limit
2
holds, i.e., (o) t ) << 1, b.) The spin 1/2 nuclei are all at equi
valent positions at a distance b from the center of a spherical mole
cule of radius a, and 3.) The motions of the molecules can be con
sidered to be rotation' and diffusion, characterized by a single cor
2
relation time % = 2a /D, where D is the translational diffusion
104
coefficient. The resulting equation is
^ ^"^^ {1 + 0.233.
T,3 T,3 5aD
^1 2
a
+ 0.15
+ •••}, (4.37)
where ri is the number of spins (protons in our case) per unit volume.
The use of this formulation in the case of NH is indeed question
able. One is concerned in particular with the use of a single cor
relation time to describe both reorientation and diffusion processes.
Perhaps a much better model for liquid NH_ would be one in which a
single correlation time describes only rotation about the symmetry
axis and another distinct correlation time describes both diffusion
and reorientation about axes perpendicular to the symmetry axis . We
will nevertheless use the Hubbard model as an approximation.
For the constant b in Eq. (4.37), we choose the distance from the
molecular center of mass to a proton. Using free molecule geometry,
we obtain a value of 0.985 % for b. The effective molecular radius,
a, which we use, is onehalf the mean distance between nearest neigh
bors, 3.56 A, determined from Xray diffraction work (43); i.e.,
a = 1.78 A. Substituting these parameters into Eq. (4.37), one
obtains ,
1 1 1.60 x 10"^ ,, „,
3= = ^ . _ (4.38)
1 2
The self diffusion constant D has been measured over the normal liquid
range by McCall , Douglass, and Anderson (70) and we will subsequently
3 3
use their .alues to calculate the T^ and T^ contributions from
Eq. (4.38).
We appeal to Abragam (7), pp. 305313 to evaluate the scalar re
4 4
laxation contributions, T and T„. Recall that' the observable in
105.
direct spinspin interaction, discussed in Sec. 2.3.4., is a scalar
interaction of the form hJI"S between the nitrogen nucleus and protons
on the same molecule. Due to the scalar nature of the interaction it
is invariant to molecular reorientation. We discussed qualitatively
in Sec. 3.4.2. the effect this interaction could have on the FID; now
we must consider how Zeeman transition inducing fluctuations may be
transmitted to proton sites via this interaction. Fluctuations at
proton sites may be produced by the indirect spinspin interaction
through two distinct physical processes: a.) Chemical exchange of
protons between neighboring molecules, and b.) Rapid fluctuation of
the nitrogen nucleus among its Zeeman states due to electric quadrupole
moment coupling to the rapidly reorienting molecular electric field
gradient. Magnetic field fluctuations by process a.) occur when the
N nuclei on two molecules are not in the same state when chemical
exchange of protons occurs between them. Abragam labels relaxation by
processes a.) and b.) respectively, scalar relaxation of the first
kind and scalar relaxation of the second kind .
Scalar relaxation of the first kind is only effective in causing
Zeeman transitions when x is within a couple of orders of magnitude
of the Larmor period. When this process is effective the proton mul
tiplet structure is of course not observable, i.e., no beat pattern is
observable in the FID although T„ may be many times J . The follow
4 4
ing equations relating T and T„ to t are derived with the assimiption
that when chemical exchange occurs it is. an instantaneous process. We
have,
1 1 J^S(S + 1) le , .
1 I S e
106
and
^2
1 J^S(S + 1)
T
T +
(4.40)
14
where S = 1, the N spin, and ca and co are respectively the Larmor
precession frequencies of the protons and the nitrogen nuclei.
Scalar relaxation of the second kind is possible when the S nucle
14
us ( N in our case) has an independent relaxation mechanism which
limits its T^ to less than J , e.g. electric quadrupolar coupled with
molecular reorientation. Again, the multiplet structure is destroyed
under these circumstances. In the short correlation time limit, where
14 14 4
T^ = T„ for the N nucleuss, the equations relating T of N to T^
and T„ of the protons are identical with Eqs. (4.39) and (4.40), but
14
with T replaced by T of N. Our nitrogen T measurements, when
used in Pople's theory (30), and high resolution CWNMR work (19,71,72),
indicate that in pure NH„ the multiplet structure is not destroyed.
Therefore, we can expect no contribution from scalar relaxation of
the second kind to proton T (substitution of nitrogen T^ values for
4 14
T in Eq. (4.39) yields values on the order of T ^ 10 seconds.)
Scalar relaxation of the second kind has a drastic effect on proton
T„, as was discussed in Sec. 3.4.2. Eq. (4.40) with x replaced by
the nitrogen spinlatt'ice relaxation time is not even approximately
, . , , ^nitrogen ^1 , , ^, ^ ^ • ^
valid when T^ < J ; whereas under the same set of circiimstances
Eq. (4.39) is a good approximation. This is a manifestation of the
widely separated regions of frequency to which the respective relaxa
tion times are sensitive. The more exact theory of Pople (30) is
4
applicable for predicting T^ under the circumstances. We note from
previous high resolution work on pure NH (19,71,72) that the multi
107
plet structure is indeed observable, but the lines are broadened,
through the indirect spinspin interaction, by electric quadrupole
induced fluctuations among the nitrogen Zeeman levels. Recall that
two distinct transverse relaxation times govern, the decay of M *(t).
One may estimate from the high resolution traces of ref. (19) that
T^ is on the order of 100 msec, roughly two orders of magnitude
less than Tf^
Although the inhomogeneity of our magnetic field limited the actual
duration of the proton FID to about 4 msec, a far too short FID to
observe the beats predicted in Sec. 3.4.2., by use of the HahnMax^^/ell
spin echo technique (16) we increased our pulse spectrometer resolution
to the degree such that a J splitting in NH_ as small as 10 Hz could
be observed. We did not, however, observe the expected beats in the
echo envelope. This was indeed a disturbing dilemma until a search of
the literature revealed an early paper by Ogg (72) describing his
difficulty in observing, by high resolution CWNIffi techniques, the ex
pected NH„ proton multiplet structure. He found that an H^O impurity
of only a few parts per million in his NH samples completely x^7ashed
out the multiplet structure and narrowed the observed singlet to a
width less than the center line of the pure sample multiplet. This
experiment indicates that H„0 is a fantastically efficient catalyst
for protonproton chemical exchange. If one could arbitrarily vary
the H2O concentration in liquid NHo while observing the true FID, one
would observe the following effects as the H„0 concentration was in
creased: a.) Pure NH„ — a FID, xjith beats of frequency J, decaying
with time constant of roughly 100 msec, b.) NH + critical concentra
■ tion of H„0 — a FID with no observable beats and T„ on the order of
108
J = 23. msec, finally, c.) NH» + high concentration of H2O— a FID
with no beats and T„ = T , where T may be affected by the exchange
process according to Eq. (4.39). Our T measurements using the un
modified CarrPurcell (67) pulse sequence are plotted as a function of
absolute temperature in Fig. 22. On the basis of our T^ measurements
and the fact that we could find no indication of multiplet splitting
in our sample, we presume that the theory presented for scalar relaxa
tion of the first kind is applicable for our sample. Before continu
2
ine. we note that since our T„ measurements indicate t ^ 10 sec,
=>» 2. e
TI^ 'J 10 sec, we may neglect the contribution of Eq. (4.39) and the
second term in brackets in Eq. (4.40). We must therefore contend only
EXP
with the following contribution to !„ :
V = 1.27 X 10^ T . (4.41)
^2
, . r^EXP , ^EXP
We have now considered all contributions to T and T„ , and
combine Eqs. (4.33), (4.35), (4.36), (4.38) and (4.41) to write the
theoretical expressions. We denote by a subscript t that these values
are theoretical. One obtains,
^=1.02x10^^ +1.60.JLi0l, (4.42)
EXP X.U^ A XU L ^
T
It
and
^ 9.87 xlO^Q T + ^^Q I ^°"% 1.27 X 10^ T . (4.43)
3XP c D e
^2t
In Sec. 4.5.1. we calculated the rotational correlation time vs.
temperature and corroborated our results with those obtained through
alternate approaches. We thus feel that the values calculated for x
14
using the solid state shifted value of the N quadrupole coupling
109
O
(M
in
ro
O
to
in
cvj
eg
LU
O 3
OJ
<
ir
LU
Q.
ID
:e
OC
LU
3
tnl
H
^UJ
^
1
_j
Hi
S
o
u
en
t
ca
(M
o
o
o
o
O
o
o
o
O
o
O)
CO
N
U)
in
^
ro
OJ
m
O
O
o
CVJ
15
o
CD
(oesuj) \
H
LU
s
LlI
q:
8)
z
8>
z
o
CE
a.
if
no
constant (3.47 MHz) may be used for t in Eqs . (4.42) and (4.43) with
out further justification. The values of the diffusion constant D
were taken from ref. (70). Fig. 23 depicts a comparison of our ex
EXP
perimental results with those of Smith and Powles (43) and the T^^
values calculated from Eq. (4.42). The top, solid curve is the cal
EXP
culated T., values. The heavy dotted straight line represents the
SmithPowles results, as well as could be determined from Fig. 1 of
ref. (43). The actual experimental scatter of our points, taken at
nine different temperatures, is represented as heavy vertical bars
through the thin dotted line, which represents a smooth curve draxra
to minimize the experimental deviations. Note that the theoretical
values predict a gentle curve, but nothing so striking as what our
data display. One might easily explain a slight difference in the
absolute value of different experimental results in terms of slight
differences in sample purity, but the large curvature of our values
is unexplainable in these terms. We note that the slope of our ex
perimental curve is approximately equal to that of the other curves
at the low temperature end but begins to rapidly decrease when the
temperature reaches approximately 210 K. This interesting behavior
has not been satisfactorily explained; possibly a clue could be ob
tained through an isothermal measurement of T vs. sam.ple pressure.
In view of the critical dependence of T on sample purity, pressure
work of the type proposed would be very difficult.
We now return to Eq. (4.43) and our experimental measurement of
T^ . Simplification of Eq. (4,43) may be achieved by noting that
FXP
1/T ^ 10 and the sum of the first two terms on the right hand side
are on the order of 0.1, a contribution less than the estimated +20%
Ill
_: J 1 1
30
20
10
9
8
7
o _
uj 6
K 5
4
BP
Theory
Ref.(34)
This work
42 4.3 44 4.5 4.6 4.7 4.8 4.9 5.0 5.1
Fig.23.L0Gn. ) vs. lOy(TEMPERATURE)
112
experimental error. If these terms are dropped and one assumes the
, standard Arrhenius form for the temperature dependence of x^,
Eq. (4.43) may be written in the familiar form,
,e
LnT^^ =A4
i
io\
^. C4.44)
suitable for fitting experimental data by linear regression. We use
•pyp EXP
our experimental values T^ for T^^ and fit to Eq. (4.44). Fig. 22
depicts our data, represented by the circles, with the solid line rep
resenting T„ . One obtains from the linear regression fit a value
of.
E^ = 0.99 + 0.32 kcal/mole, (4.45)
for the protonproton chemical exchange activation energy. The error
limits represent a linear regression estimate of the 80% confidence
interval. This value is considerably lower than one might expect from
previous work (19) , which employs KNH rather than HO as a catalyst
for protonproton exchange. They find the chemical exchange activation
energy to be independent of BCNH„ concentration and equal to 4.0 + 0.5
kcal/mole. It is conceivable that the protonproton exchange activa
tion energy may be a function of catalyst. In light of the tremendous
efficiency with which H„0 impurity in NH destroys the pro.ton multi
plet structure, Eq. (4.43) may well be a reasonable estimate.
CHAPTER V
INSTRUMENTATION AND MEASUREMENT PROCEDURE
5.1. Design of Cryostat and Sample Handling Procedure
5.1.1. Sample History and Handling; Procedure
The ammonia sample used for all experiments came from, a single
source: The Matheson Co. It was analyzed by J. Catesel of the
Matheson Co. and found to be 99.999% pure NH . A first prerequisite
for design of the sample handling system was the preservation of sample
purity. In particular, oxygen contamination was to be avoided. The
handling of ammonia presented somewhat of a problem in that it is quite
toxic and chemically active. Dr. J. Kronsbein (73) advised the use of
only ferrous metals for handling the gas. The sample chamber required
use of a nonconductive, nonmagnetic material. We chose the polymer of
trlfluorochloroethylene, known by the trade name of KelF, for the
sample chamber primarily for its inertness and desirable low tempera
ture properties.
A diagram of the gas handling system is shotm in Fig. 24. It is
constructed of only ferrous metals, teflon and KelF. All joints are
either welded, metaltometal sealed, or sealed using teflon. The
system xjas thoroughly cleaned, assembled, and checked for leaks using
a Veeco MS9AB mass spectrometer leak detector. The complete system
was then connected to a standard design coldtrapped diffusion vacuum
pump and pumped out to roughly 10 Torr. Outgassing was accomplished
113
114
Valves
To Vacuum System
Steel Tube
99.999 7o Pure NH,
400 in^ Low Pressure
Storage Cylinder
•^
cr^
Stainless
Steel Tube
Cryostat
KeiF
Sample
Chamber
Fig. 24. DIAGRAM OF THE GAS HANDLING SYSTEM
115
by heating the system to above 370 K using a heat gun. This process
was repeated until no increase in pressure was observed upon heating.
The pumpout procedure was carried out over a period of several days.
Ammonia gas was introduced into the system at a pressure such that
when the sample chamber was filled with condensed liquid NH„, a pres
2
sure of approximately 5 lb/in above atmospheric remained in the low
pressure system. The sample system circuit remained intact throughout
the experiment.
The first sample remained in the cryostat and low pressure system
from 10 May 1968 until 10 June 1958. During the course of this period,
the sample was condensed into the sample chamber and boiled off a num
ber of times. The first sample was then pumped out and the system
outgassed, and a second sample was introduced and condensed into the
sample chamber. Six runs were made on Sample II, measuring proton T..
vs. temperature in the liquid. Consistent results were obtained, but
the value of T^ at 195 K was consistently about 1.5 sec lower than
that of ref. (43), indicating a slight impurity. The sample was
changed again and further proton T^ measurements made. It was found
that if the sample remained in the liquid state over a period of time
(roughly three days), the proton T values increased slightly, reach
ing a limiting value of — 5 sec at 198 K. A temperature run was made
on sample III (proton T vs. temperature) after the limiting value
\ was reached, then the sample was again changed. Proton T values
remained invariant, within experimental error of approximately +10%,
to both the amount of time the sample remained in the liquid state
and a change of the external magnetic field. The proton T data in
solid NH were taken using sample IV, then confirmed by changing
116
samples again and taking spot points throughout the temperature range.
Perfectly consistent results were obtained from both samples in the
solid.
5.1.2. Construction and Performance of the Temperature Control Cryostat
The temperature control cryostat was designed to be as versatile as
possible, and is useful in the temperature region from 4.2 to 200 K.
The design is such that complete disassembly is easily effected in
order that repairs or modifications may be made with a minimum of dox.m
time. It would be quite feasible to design a number of interchangea
ble probe sections for use in different experiments. This disserta
tion is obviously no place for detailed design drawings, but Fig. 25
will Illustrate the basic design details.
The control section is machined from a solid piece of aluminum.
All tubes connecting the probe to the control section are ten mil wall,
nonmagnetic stainless steel. The vacuum space between the two outer
tubes prevents the sample from solidifying and plugging the sample
tube. He exchange gas occupies the space between the sample tube
and inner vacuum jacket tube. The top of the brass outer can is seal
ed with a Wood's metal solder joint. The temperature bomb was ma
chined from a single cylinder of oxygenfree highconductivity copper;
its final mass is 909 grams. Six 1/4 x 20 brass bolts fasten the
temperature bomb, and a vacuum tight seal is effected using an indium
0ring. A three mil brass shimstock radiation shield radiatively
decouples the temperature bomb from the outer brass can. Convective
heat transfer betx^?een the temperature bornh and the outer can may be
4
controlled at will by varying the pressure of a He exchange gas in
side the outer can. Heat transfer betv7een the sample and the
117
37" 30.1" lir
i3.r
ya'od  io MIL U.PIUI.
i/Z* oj ' to Mil. Viftui. OT^^/VtffiS
/«/« lAttiuM Jacket 'TZ^e
o
OoTSIOC. lAiUWr JicUiT TuoS
^<uuM JrtCKir
3 Mil. ^/fflJS ^r^'^JToc*;
Bkasj Vacuum Jfi^CK^r C.Ar
/r /Pis/trAf^a /He/ii<'K^fi£Teix
OFHC Ca/vsn Bom a
Fig. 25. THE TEMPERATURE CONTROL CRYOSTAT
118
temperature bomb is maintained both by thermally grounding the coil
4
surrounding the sample chamber and by providing a He exchange gas.
Temperature control is effected by first surrounding the outer can
with a slowly boiling cryogenic liquid, and then allowing heat trans
4
fer between the temperature bomb and the cryogenic bath, via a He
exchange gas, until the desired sample temperature is reached. One
3
then evacuates the outer can to a pressure ^ 10 Torr and supplies
energy to the temperature bomb, via a heater, at the same rate that
energy is being lost to the cryogenic bath. The bomb heater was
wound in two sections, each covering approximately one inch of the
bomb length, positioned in such a manner that the center of the sample
chamber lies between the coils. The coils were wound independently
so that a thermal gradient over the sample could be cancelled by
varying the relative power input to the coils (see Fig. 28). Each of
the coils consisted of 700 Ohms of teflon coated, coppernickel alloy
wound noninductively around the bomb.
Electrical leads from the control head into the probe section are
as follows: 1.) Three //32 varnished copper wire leads for the heat
ers — electrical resistance of each is 0.6 Ohms, 2.) Four #36 varnish
ed copper wire leads for the platinum resistance thermometer — elec
trical resistance of each is 1.4 Q, 3.) Tv70 three mil copper wires
for a thermocouple pair and 4.) The rf lead of rg/u 178b shielded
50 Ohm cable. The rf lead was passed between the 1/8 inch stainless
sample tube and the inner vacuum jacket tube; the shield was grounded
to the temperature bomb. All remaining electrical leads were chan
nelled do^ra a 1/4 inch id. stainless steel tube and thermally grounded
to the outer brass can. Apiezon "N" grease was used to provide good
119
thermal grounding (74) between the heater and the temperature bomb,
and the electrical leads and the brass can.
5.1.3. Temperature Measurement, Contr o l and Stability
Temperature measurement and control were effected through the use
of a Leeds & Northrup (L & N) type 8164 fourlead platinum resistance
thermometer which was calibrated by the National Bureau of Standards
(Test #037794). The actual thermometer used was L & N serial #1692601.
Through use of an L & N type G1 Mueller Bridge and a type 8068 mer
cury commutator, the absolute thermometer resistance was measured
irrespective of lead resistance. We used a Keithley model 149 milli
microvoltmeter as a null detector.
The thermometer, a cylindrical device 6 cm long with an o.d. of
5.5 mm, was mounted in a hole drilled into the side of the bomb par
allel to its axis. Care was taken to coat the thermometer over its
complete length with Apiezon "N" grease in order that good thermal
contact was established. Care was also necessary when soldering the
thermometer leads as it is advantageous to minimize both thermal emf,
caused by poor solder joints, and the difference in lead resistance of
the two lead pairs. Good quality tinlead solder was found to be much
more satisfactory than a "low thermal emf ", solder (this perhaps re
flects the author's inability to obtain a decent solder joint with the
latter).
Thermozeter resistance decreases from 25.4800 Abs. Ohms at the
triple point of water (273.16 K) to 1.91969 Abs. Ohms at 50 K. Fur
thermore, the resolution (dR/dT) decreases from 0.1 Ohm/K at 273.15 K
to 0.094 Ohm/K' at 50 K. The resolution becomes very poor at tempera
tures below 20 K, and the thermometer is almost unusable below 10 K.
120
Throughout the region of temperature in which measurements were made,
we could easily resolve temperature changes of +5 mK corresponding to
a resistance change of 5 x 10~ Ohms. We estimate our absolute tem
peratures to be accurate to +0.1 K and our temperature control to be
stable to better than one part in ten thousand.
The operational principle of our temperature control unit is
straightforward: For a given set of conditions (cryogenic bath tem
perature, desired sample temperature, pressure in cryostat vacuum
jacket, etc.), the power input to the bomb must equal the energy lost
per unit time to the cryogenic bath, if one expects an equilibrium
situation at a desired temperature. Operationally, one sets the
Mueller bridge to the resistance corresponding to the desired bomb
temperature; if the bridge is out of balance an error voltage is in
dicated by the Keithley model 149. The magnitude of the error voltage
is a function of the amount of bridge outofbalance and the voltage
gain setting of the Keithley. The polarity of the error voltage is
determined by whether the bomb is too hot or too cold relative to the
desired temperature. If the bomb is too cool, one increases the power
input until a steady state condition at the desired temperature is
reached; if too hot, the power input is decreased. The power input is
a quadratic function of the voltage applied across the bom:b heaters.
A perfect steady state condition at a constant input power cannot be
conveniently met because slowly changing conditions of the system
(vacuum fluctuations, power supply drift, etc.) continuously demand
slightly different input power. We provide this slight variation in
input power by applying the error voltage developed by the Keithley to
a programmable pov;er supply. The error voltage developed (10 to to
121
+10 Vdc) produces roughly an equal change in voltage across the bomb
heater. Thus the change in power input is proportional to the square
of the outofbalance signal. This quadratic behavior provides ex
ceptionally good temperature control.
The temperature control loop is illustrated in Fig. 26. All con
trol is derived from a programmable KrohnHite model UHRT361R power
supply. A Heathkit model PS4 regulated power supply is not a part of
the control loop and is used as a booster to facilitate rapid changes
of the bomb temperature. The gain of the loop is determined by the
sensitivity setting of the Keithley. The temperature stability fig
ures presented earlier are based on a sensitivity of 1 yV full scale
with 6 V applied to the Mueller bridge. No loop instability was ever
observed; this was no doubt due to the low thermal resistivity and
high thermal capacity of the bomb, and the relatively small available
power change.
The average power input to hold a given temperature under typical
conditions is of some interest both because it is a measure of the
power input to the cryogenic bath and because it facilitates setting
the average heater bias voltage. Fig. 27 illustrates the average
power input to hold a constant temperature vs. the absolute tempera
ture, for the case of a slowly boiling liquid nitrogen bath. The
4
pressure in the cryostat vacuum jacket was approximately 10 Torr
and was held constant by continuous pumping. The two heaters were
connected in series for this measurement. Notice that a +007 watt
control, corresponding to our available +10 V control, has a steady
state effect of +12.5 K at 90 K but only a + 7.25 K effect at 200 K.
We mentioned previously that the heater coils were wound such that
122
Heothkit PS 4
KrohnHite UHRT36IR
Keithley model 149
Millimicrovoltmeter
LSN Type Gl
Mueller Bridge
>,^^
L&N Type 8154
R. Thermometer
OFHC Copper
Temperoture Bomb'
Heoter Coils
J
■A.
Fig. 26. BLOCK DIAGRAM OF THE TEMPERATURE CONTROL LOOP
123
CD
CO
f~ tD ih ■^_ to
(SliVM) indN! yBMOd
cvj —
^
en
124.
any temperature gradient existing in the sample chamber could be can
celled. In order to determ.ine if this procedure would be necessary,
the temperature gradient was measured between the top and bottom of
both the sample chamber and the outside of the temperature bomb, for
the case of a steady state temperature established by a current flow
ing through the series connected heaters. The measurements were made
using a gold~2.1 atomic % cobalt— copper thermocouple pair placed in
the positions aa and bb, as depicted in Fig. 28. The outside
Stoinless Sample Tube
R. Thermometer Hole
Heater Coil
■KelF Sample Chomber
OFHC Copper Bomb
He Exchange Gas
k
l.so
Fig. 28. SAMPLE CHAMBER 8 TEMPERATURE BOMB
125
thermocouple pair (aa) was thermally grounded to the respective ends
of the bomb via mica insulation and Apiezon "N" grease; measurements
4
were made with the vacuum jacket evacuated to roughly 10 Torr. The
inside thermocouple pair was attached to copper disks, 60 mils thick
and 450 mils in diameter, separated by a two inch styrofoam spacer and
positioned inside the bomb as shown in Fig. 28. Throughout the meas
4
urements, a He pressure of one atm. was present inside the bomb.
Thermopotential was measured in each case after thermal equilibrium
was established relative to a liquid nitrogen bath. The actual meas
urements were m.ade with the Keithley model 149 millimicrovoltmeter .
In order that any extraneous thermopotential generated at the Keithley
input be cancelled, a thermal free reversing switch was used; the
average of the forward and backward voltages was taken as the true
value. Fig. 29 illustrates the results of the measurements plotted
as the difference in temperature of the respective thermocouples vs.
the steady state temperature. The thermocouple calibration tables
used vrere those of Powell, et al (75). We consider these results to
be excellent justification to ignore the problem of thermal gradients
across our sample when the heaters are connected in series; thus the
series connection was used for all experiments.
5.2. Measurement Procedures Used
5.2.1. Measure ment of T by 90°t90° Pulse Sequence
This measurement procedure is quite straightforx>7ard for the case
of the existence of a single spinlattice relaxation time. We have
illustrated from various theoretical results that the existence of
a single T^ governing the relaxation of protons in NH„ is a good
126
ACROSS SAMPLE (K)
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127
approximation over most of the region studied. If one makes this as
sumption, Eq. (3.69) may be used to measure T . The initial conditions
are established by applying a 90° pulse, i.e., M (t=0) = 0. After the
90° pulse one observes a FID, the height of which, immediately follow
ing the pulse, is proportional to the thermal equilibrium magnetiza
tion M . In practice however one cannot observe the FID at t=0 due to
o
the intrinsic deadtime of the electronics. A long deadtime may com
pletely obscure a short FID; we consider this problem in more detail
later. In practice one chooses a point at time t from the initiation
of the 90° pulse (a point such that the FID is observable) , and makes
all amplitude measurements at this point, keeping t constant. If one
applies a second 90° pulse a time t < T after the first, the ampli
tude at t + T will be smaller because thermal equilibrium has not
o
been reestablished. We associate the amplitude V, (0) obtained in
this manner after the first pulse, with Mq ; likewise, if the 90° pulses
are t apart, we associate V (t) with M2(t). One may then write
Eq. (3.69) as,
Vd(T) = Vd(0)[l  EXP(t/T^)]. (5.1)
The two 90° pulses, a time t apart, are referred to as a single
90°T90° pulse sequence.
If several 90°t^90° pulse sequences are applied with t different
in each case, one may plot the recovery envelope if the time between
each pulse sequence is greater than 5 T . Fig. 30 illustrates the
procedure described above.
The spin lattice relaxation time T^ is extracted' from the experi
mental data using Eq. (5.1). One plots Ln[l  V^^ (t)/V^ (0) ] vs. t
128.
and finds
T^ =  1/m,
(5.2)
where m is the slope of the straight line. This procedure has the
advantage of illustrating immediately: (1.) The precision of the
90° pulse tuning and (2.) any nonlinearity in the recovery. If the
90° pulse is tuned correctly the straight line should pass through the
origin, if not the intercept will be finite. One should be particu
larly dubious of any T value obtained when the straight line inter
cept is positive.
Recovery Envelope
'90' Pulse
— H H — Amplifier Dead Time
90' Pulse
TIME
Fig. 30. THE 90* 1  90' PULSE SEQUENCE
Another procedure used to obtain T, is that whereby one simply
seeks the pulse separation t which makes the quantity Ln(l  V,(t)/V, (0))
d d
equal to minus one. At this point, clearly '^ equals T^ . This proce
dure may lead to large errors if the 90° pulse tuning is not correct.
The procedure used in this work is the linear regression technique.
The experimental data were fit to a straight line of the form,
Ln(l  V,(t)/V^(0)) = mi + b. (5.3)
d d
129.
This procedure has the advantage of removing random errors in a sys
tematic manner. One may also obtain an estimate of the variance of
the linear regression coefficient m from which one may estimate the
statistical interval of confidence of T^ . The intercept b, from
which one may estimate the accuracy of one's 90" pulses, is also
available. This procedure is of course entirely too laborious with
out the assistance of a small computer. We found the HewlettPackard
9100A ideal for this purpose. Clearly this procedure would be xvell
suited for an online computer fed by a boxcar integrator. All sta
tistical procedures used were taken from Ostle's book (76) on the
subject.
The actual data were taken using the beam intensification feature
of the Tektronix 545 oscilloscope. This procedure is identical to
that explained by De Reggi (77) and thus will not be reiterated here.
5.2.2. The Measurement of the SpinSpin Relaxation Time
As motivation for using these techniques, we first discuss the
effect of magnet inhomogeneity on T„ of liquids. In Sec. 3.3.2. we
considered the relationship between the lineshape and the FID when
our external field H was perfectly homogeneous, but this is never the
case. Inhomogeneities in real magnets are primarily the result of two
factors: 1.) The finite diameter of the pole faces leads to a radi
al field gradient; the field is largest at the center and decreases
as a function of radial distance. 2.) A very serious form of in
homogeneity results from different regions of the pole faces magnetiz
ing to different values of H for a given constant magnetizing force.
The latter problem should not occur in a welldesigned magnet, but
the first is an intrinsic property. One widely used scheme of reducing
130
the radial gradient over an area at the center of the pole faces is to
provide a ridge of increased thickness around the outside edge. The
width of the raised ridge and its thickness are used as parameters to
adjust the homogeneity of the central portion of the field. The pole
faces on our magnet are not of this design and the radial gradient is
approximately 50 mG/cm at H = 7.05 kG over a circular area roughly
two centimeters in diameter. One may, incidentally, improve the ef
fective homogeneity of a flat pole face magnet by the following very
simple operation: Run the field approximately 100 G above the desired
field, then reduce it to the desired value.
For protons, a difference in local field of 50 mG at 7.05 kG cor
responds to a difference in precession frequency of roughly 2.1 kHz.
Since the natural linewidth of most liquids is in the range of 0.1 to
10 Hz, clearly the lineshape is characteristic only of the magnet.
We wish to consider the effect of the magnetic field inhomogeneity
and sample geometry on the FID, for the limiting case where the natu
ral lineshape f(v) may be approximated by a delta function. In this
case, as we showed in Sec. 3.4.2., the magnetic field distribution
function over the sample takes the place of the lineshape function
in Eq. (3.29) and we have
00
V,(t) = a^ /g(v)cos27TVtdv. (5.4)
—CO
The magnetic field distribution function g(v) is written in terms of
the distribution of Larmor frequencies it produces over the spin en
semble, as seen in a frame rotating at v . The problem is simplified
greatly by choosing one's laboratory coordinate frame and sample ge
ometry in such a manner as to maximize symmetry of the problem. In
131.
practice this is usually done by simply centering a symmetric sample
chamber over the center of the magnet pole faces.
We consider the experimental case that we were presented with:
That of a long, thin cylindrical sample chamber, of length i and
radius a, geometrically centered relative to the magnet pole faces.
We use the coordinate system shown in Fig. 31(a.); and since a << Ji
we may approximate the magnetic field seen by the sample as
H = k(H^  Gy), (5.5)
where G is the radial field gradient constant. Note from Eq. (5.5)
that both +y halves of the sample 'see' identical field distributions.
If we choose v = yH /2it, g(v) becomes
g(v) =
4tt yG&
yG2, ^ ^ 4tt
(5.6)
v>^
4Tr
Substitution into Eq. (5.4) gives,
V^(t) = a^ sin(y £Gt/2) ^ ^^^^^
We plot values calculated from Eq. (5.7) in Fig. 31 (b.) using '
G = 8 mG/cm, £ = 5 cm, y  2Tr x 4.26 x 10 and a„ = 1. For compari
son, a FID from our second NH„ sample is normalized and plotted on
the same graph. Clearly the experimental FID is more highly damped
than the theoretical curve, no doubt due to the indirect spinspin
interaction. The important point is that the oscillation is present,
and explainable in terms of a not unreasonable gradient across the
sample chamber. We used the technique mentioned previously to maxi
mize the field homogeneity.
132
/Sample Chamber
Mognet Pole Face
Rg. 3l(a.). SAMPLE CHAMBER 8 MAGNET COORDINATE SYSTEM
Experimental points
Absolute value of Eq.(5,7)
2 4 6
^^^c
10 12 14
t (msec)
18 20
Fig. 3Ub}. RELATIVE FID AMPLITUDE! PROTONS NH^) vs. TIME
133
In the literature one finds mentioned an apparent transverse relax
ation time T„. T' is the effective transverse relaxation time that
one associates with the observed decay of the FID after a single 90°
pulse. In the case of an oscillatory decay one might define T„ in
terms of the envelope of the oscillatory maxima. This quantity is
tenuous, at best, but useful, primarily as a parameter by which to
gauge the homogeneity of one's magnetic field. For the case of most
solids, T„ is meaningless because the dipcle coupling in the solid
causes such rapid damping of the FID that one observes essentially the
true T„, assioming that a magnet of reasonably good quality is used.
In light of the preceding analysis one could wonder how the
intrinsic liquid sample T2 may be measured when one is forced to work
with real magnets. The elegant spinecho technique may be used to
eliminate the effects of magnetic field inhomogeneity on measurements
of T„ . Hahn introduced the spinecho technique in a classic, very
lucid paper (78) concerning the response of the spin system to a
90''T90'' pulse sequence, when T* < t < T^. He finds that, although
the spins fan out rapidly in an inhomogeneous field after a 90° pulse,
the spin ensemble may be refocused by a second 90° pulse at a time x,
subsequent to the first, if each spin 'sees' relatively the same
field at times t=0 and x. If x meets this condition, the refocused
spin ensemble creates a maximum in the nuclear signal at a time t
after the second 90° pulse. This maximum is called a 'spinecho.'
The spin echo is maximized by a 90°x180° pulse sequence as is ex
plained in the very readable CarrPurcell paper (67).. Any process
which tends to change the local nuclear fields over the period x will
reduce the echo amplitude. The process of selfdiffusion in an
134
inhomogeneous magnetic field is particularly effective in reducing the
echo amplitude. Hahn finds that the echo envelope, obtained by plot
3
ting echo maxima for different x, is modulated by the factor EXP(t k) ,
where k is proportional to the product of the square of the field gra
dient with the molecular selfdiffusion constant.
The CarrPurcell technique tends to eliminate the damping effect
of diffusion through an inhomogeneous field by applying a series of
180° pulses after the single 90° pulse (see Fig. 34). They find the
echo envelope to be described by
V^(t) = V^(0)EXP[(t/T2) + (yVoT^/lZm^)], (5.8)
where D is the atomic selfdiffusion constant, G is the field gradient
constant and m is the number of 180° pulses applied over the observed
period. This procedure has the effect of refocusing the precessing
spin ensemble at such a high rate that spins are kept in the observ
able ensemble even though selfdiffusion through an inhomogeneous
field is continuously, but slowly, changing the local fields of all
spins. Diffusion damping of the envelope is made negligible by making
m large enough. i
The CarrPurcell technique is particularly convenient because one
may photograph the complete echo train displayed on the scope in a
matter of seconds, whereas using the Hahn technique one must wait for
the sample to reestablish thermal equilibrixim following each 90°t90°
pulse sequence. One serious drawback is that the 180° pulse tuning is
very critical because a small error is amplified m times over the
course of the measurement. This leads to oscillatory echo envelopes
which may lead to errors in T„. One should plot Ln[V (t)/V (O)] vs. t to
135
confirm a single exponential echo envelope. A modification of the
CarrPurcell technique by Meiboom and Gill (79) reduces the sensitivity
to 180° pulse tuning by shifting the phase of the 180° pulses relative
to that of the 90° pulse by tt/2.
5.2.3. The Measurement of Linewidth and Second Moment
Discussions of continuous wave measurement technique abound in the
literature (see Abragam (7) or Andrew (8)), and we will not add to the
surplus here, other than with a few brief comments of how the measure
ments were actually performed and data analyzed. Our equipment will
be described in subsequent sections; suffice it to say that CW absorp
tion lines were obtained with a frequencyswept oscillator and magnetic
field modulation was used. Typical lock in detection was used giving
one, as an output, the first derivative of the absorption line with
respect to frequency.
The linewidth, plotted in Fig. 18, is the halfwidth in frequency
units between the points of Inflection (maxima and minima on the deri
vative curves) of the absorption curve. The second moment was cal
culated numerically, using the trapezoidal rule, directly from the
derivative of the absorption curve. Each side of the line was cal
culated using the relationship,
A.
M^^ = 1 i
3
(5.9)
Kf'((^.)
1 ^
where F' (co) represents the derivative of the unnormalized absorption
1
curve, which is centered at w=0. Each error bar in Fig. 19 represents
the difference 'in the values of M^ found for the respective sides of
i
the lines .
136.
All values of M„ calculated were corrected for magnetic modulation
amplitude. The modulation frequencies used, 25 or 100 Hz, lead to a
completely trivial frequency modulation correction to the second mo
ment. The correction formula presented in the equation below is a
generalization by Halbach (80) of that due to Andrew (81) . We include
the modulation frequency term, although it is negligible in our case.
»,corrected ,,EXP 1 m
^2 = ^2  IT
 f H^ (5.10)
4 m
The terms u and H in Eq. (5.10) are, respectively, the modulation
frequency and onehalf the peaktopeak modulation amplitude. Modula
tion amplitudes were measured directly at the sample chamber.
Great difficulty was experienced in obtaining a good signal in the
region for 10 to 60 K. Presumably the increase in electrical conduc
tivity of copper in this temperature region leads to nearly complete
exclusion of the alternating modulation field, even at a modulation
frequency as low as 25 Hz .
5.3. Electronic Apparatus
5,3.1. Pulsed Spectrometer
De Reggi (77) has presented quite a complete summary of what one
desires of a pulse spectrometer. We will not repeat his summary;
Instead, we launch directly into a description of the apparatus
actually used for the experiment. Fig. 32 depicts in block form the
I
pulse apparatus. i
We discussed the resonance phenomenon in Sec. 3.3.2., and pointed
out that application of a 90° pulse required application of an intense
rf voltage to the sample coil for a time to? which is inversely
137 .
proportional to the amplitude of the rf voltage. Continuous wave rf
is supplied at a constant frequency and amplitude of about one volt
peaktopeak by the oscillator. The frequency of the rf should be
approximately equal to that of the absorption line center frequency.
The gating circuit, actually an integral part of the pulsed amplifier,
will not connect the oscillator and power amplifier until a gating
pulse is applied. While a gate pulse is applied, the power amplifier
amplifies the one volt peaktopeak input, and this rf pulse is applied
to the sample coil via a "black box" coupling device. After a pulse
the power amplifier is decoupled from the sample, and the nuclear sig
nal signal, induced in the sample coil, is amplified, detected and re
corded. The timing and pulse generating circuit provides the desired
sequence of gating pulses, which are subsequently transformed into the
desired sequence 90° rf pulses.
Goted rf Power Amp.
CVV rf OsciJlator
r 1
y .
Timing B Pulse
Generating Circuits
Sample
Coil
Coupling
Network
:on^
Magnet
Sample Coil'^^^^f ^
%.
Triggered
Recording
Device
t PA^e ct or _&_ Ppst_ Amp.
jr.
High Gain ( lOOdb)
rf Voltage Amp.
Fig. 32. BLOCK DIAGRAM OF THE PULSE SPECTROMETER ELECTRONICS
138
Continuous Wave Oscillator
A General Radio Co. Type 805D standard signal generator was used
to supply carrier rf. The frequency output of this unit is continuous
ly variable from 16 kHz to 50 MHz. We measured the oscillator short
/I
term frequency stability to be +1 part in roughly 10 and the long
term frequency stability to be roughly +50 parts in 10 over a 24 hour
period. All frequency measurements x^ere made using a HewlettPackard
5245L frequency counter.
Gated Power Amplifier
The gated amplifier used is a modified Arenberg Ultrasonic Labora
tor. Inc. Model PG550C pulsed oscillator. The unit was factory
modified to perform as a gated amplifier rather than gated oscillator.
Two additional stages of amplification were added, one gated, to in
crease carrier suppression and accommodate the rf input from the Gen
eral Radio type 805D standard signal generator. This unit is capable
of applying 300 V peaktopeak pulses into a 50 ohm load. It is
capable of pulses of maximum length roughly 200 ysec without appreci
able droop. Tx'7elve output coils provide a tuning range from 0.5 to
60 MHz. We found that the coil marked for a specific frequency range
did not always give one the maximum output in this range.
De Reggi (77) mentions use of an additional gated low power ampli
fication stage to improve carrier suppression. It was our experience
that this additional stage was unnecessary with the suppression now
available through the use of low capacitance, high switching speed
diodes designed for use in computer circuits. We use two crossed
pairs of Falrchild 1N3600 diodes connected in series with the poxjer
amplifier output. These diodes each have a capacitance of 2.5 pF;
139
Master
Clock
TEK
62
Waveform
Generator
J
A
Recording
Device
/
TEK 163
Pulse
Generator
TEK 163
Pulse
Generator
Pulse
Mixer S
Selector
Ref. (82)
TEK 163
Pulse
Generator
Pulse
Mixer 6
Selector
Ref. (82)
Wong 612 AT
Pulse
Generator
Pulse
Amp.
Ref(77:
To rf Power
Amp.
Fig. 33(g.). CONFIGURATION USED FOR 90' T 90* PULSE SEQUENCE
Baseline "*~
_ 1 a p. 1 J 4 1 » . > . s J *> p = [ >
1 j . ^ .^
a
\
a
t
t. T^
Fig. 33 (b.). TYPICAL DATA OUTPUT FROM 90T90° EXPERIMENT
140.
thus this arrangement presents a series impedance of roughly three
thousand Ohms to a 20.8 MHz carrier, when they are not in a conducting
state, and a very lox/ series impedance during application of a pulse.
Pulse Program Circuitry
Even for an experiment as uncomplicated as a 90°t90'' pulse
sequence, one must contend with three distince time scales: 1.) The
length of a 90° pulse, 2.) The time t between 90° pulses and 3.) The
time interval between 90°t90° pulse sequences. The interval between
90°T90° pulse sequences, during which the spin system reestablishes
thermal equilibrium, should be no shorter than five times T .
Our pulse program circuitry is for the most part of commercial
design and construction. The units are designed to be highly versa
tile and provide one with the facility for a large array of pulse
programs. In lieu of a list of the units, we illustrate the procedures
used to obtain 90°T90° and CarrPurcell pulse sequences. Fig. 33 (a.)
illustrates the configuration used for generating a 90°t90° sequence.
Obviously it is not unique, but was found to be a highly serviceable
configuration. We found that generation of all pulses of a given
width by a single pulse generator gave one a more stable sequence than
using separately tuned units.
The Wang 612AT programmed pulse generator is the heart of the
system. This unit is capable of generating a sequence of from one to
twelve equally spaced identical pulses plus from one to twelve equally
spaced identical pulses, delayed from the first set. The width of
the direct and delayed sets is independently variable over the contin
uous range from 1 to 10 usee, and the delay time is also continuously
variable over this same range. This unit is particularly useful for
141
measuring T values of less than one second because one may increase
T in a 90''T90° pulse sequence by eleven equally spaced Increments by
simply sliding two switches. The master clock controls initiation of
the complete 90''t90° pulse sequence; it is set to initiate a new
 90°T90° sequence each 5 T sec. One may trigger the recording device
by any or all of the pulser imits.
For T < 1 sec, we use a Tek 162 waveform generator as a clock
source. It is set to repeat the 90''t90° sequence each 5 T sec. Only
one of the Tek 163 pulse generators is triggered, and it in turn trig
gers the Wang 612AT which generates the 90°t90° sequence. A Tek
tronix type 585A oscilloscope is triggered by each 90° pulse from the
612AT, and a segment of the FID after each of the pulses is photograph
ed. The segment photographed is an intensified portion of the trace,
and it always has the same temporal relationship with the leading edge
of a 90° pulse. For each value of t, a minimum of three 90°t90°
pulse sequences was photographed, and the number of signal averaging
from this procedure. Fig. 33(b.) illustrates the typical data output.
For 1 sec < T^ < 10 sec, two Tek 163 pulse generators were used to
generate the 90°t90° timing sequence. One unit was triggered at
the leading edge and the other, near the end of a sawtooth oscillation,
provided by a Tek 162, The pulse from each Tek 163 then triggers a
single pulse from the Wang 612AT. The time t between 90° pulses in
J the sequence is determined by the length of the sawtooth waveform from
the Tek 162, w^hich is triggered manually.
Fig. 34 (a.) illustrates the configuration used for a CarrPurcell
sequence. The actual pulse sequence obtained is illustrated in
Fig. 34 (b.), and 34 (c.) depicts a typical echo envelope as photographed
142
;
Master
Clock
TEK 163
®
Pulse
Mixer a
Selector
i
Recording
Device
Pulse
Mixer a
Selector
@
1
Delayed Gate
(D
TEK 162
D
TEK 163
Generator
ng.34(a). CONFIGURATION USED FOR CARRPURCELL SEQUENCE
®
5)'
>5T.
y
/
/
Gotes TEK 162
Alt i.^o° P>*!«s
@
Fig. 34 (b.) OUTPUT AT POINTS INDICATED IN Fig. 34 (o.)
^1
\l\h,
M
'^^^
fi^iihi

j.j
m\j\
%^
^
\AA,'\A
l'v\/v
w\^
T = 2msec
Fig.34(c.) TYPICAL DATA OUTPUT FROM . CARR PURCELL SEQUENCE
1A3
from the oscilloscope. The Princeton Applied Research, Inc. waveform
eductor was used primarily as a delayed gate to gate the Tek 162 on
for a predetermined time, during which a number of 180° pulses from a
Tek 163 Xv^ere generated. The number of 180° pulses generated during
the gate pulse is governed by the sawtooth repetition rate set on the
Tek 162. Figs. 34 (a.) and (b . ) are selfexplanatory so we will not
belabor the reader with additional words of explanation.
Data Recording Device
All data were recorded on type 47 polaroid film from the screen of
a Tektronix type 585A oscilloscope using a type C12 Textronix trace
recording camera. A tektronix type 86 plugin preamplifier accepted
the output of our receiver.
Magnet
The magnet used was a 12 inch Varian type V40123B, fitted with
factory supplied cylindrical pole caps which provide a 3.5 inch air
gap. With this pole cap configuration, the maximum obtainable field
is approximately 9 kG; and the factory specified minimimi field homo
geneity is roughly 50 mG/cm over a circular area 2 cm in diameter.
Frequency and Time Measurements
All frequency and time measurements were made using a Hewlett
Packard model 5245L counter equipped with a type 5262A time interval
unit. These units allowed very precise measurement of both the time
interval between pulses and pulse width.
The Coupling Network and Receiver
The coupling network and receiver are considered together because
they are so intimately connected. More properly, perhaps, one should
divide this discussion into three distinct sections. Taken in order
144
from the sample coil these are: a.) The coupling network and rf pre
amplifier, b.) the wideband rf amplifier and c.) the detector cir
cuitry and postaraplifier. The coupling network and preamplifier are
burdened with a most difficult task: That of coupling a '^ 300 V rf
pulse from the power amplifier to the sample coil during a pulse, then
recovering rapidly from the pulse to couple a high gain amplifier to
the sample coil in short enough time to observe a FID, which may be
less than 20 ysec long and have an amplitude at the coil of only a few
tens of a microvolt. The wideband amplifier must provide sufficient
voltage gain, as supplement to that of the preamplifier, to render the
nuclear signal visible. The detector removes the information carried
by the amplitude modulated rf carrier, and the postaraplifier primarily
isolates the detector circuit from recording devices.
The requirements placed on the detector and postampllfier are
quite stringent. The detector must accept the amplitude modulated
carrier, modulated by the lineshape function expressed in the time
domain, and remove the information in such a manner as to introduce
no distortion. The detection process is basically a rectifying and
filtering process whereby, ideally, the detector is perfectly linear
and the filter passes only frequency components contained in the FID.
Normal diode detection is only suitable if the diodes are forward
biased into a region of high linearity. This is done most effectively
through the use of phase coherent detection [see the following re
ferences, taken at random from the literature: (83), (84), (85), and
(86)], which has the added advantage of an intrinsic increase in the
observed signal to noise ratio. Ref. (86) presents a very good
discussion of phase coherent detection. Unfortunately, our General
145
Radio Co. type 805D signal generator lacks the short term stability
necessary for phase coherent detection; therefore, we were constricted
to use of diode rectification. Distortion will, of course, be intro
duced if the detector and postamplifier have too narrow a bandwidth
to pass all the Fourier frequency components present in the FID.
The requirements placed on the wideband amplifier are similar to
those above. It must linearly amplify the rf and have a bandwidth,
about the central Larmor frequency, wide enough to accept all frequency
components present in the FID. Ideally, the bandwidth of this ampli
fier is such that it will provide roughly 80 db gain over the total
range of frequency at which one may wish to work.
For our work in liquid NH„, we used a Lei, Inc. model 5182 ampli
fier, factory modified to provide roughly 120 db overall gain at a
center frequency of 3 MHz. The bandvridth of this unit is '~ 1 MHz.
Modifications of the input stage will be described subsequently. We
utilized an Arenberg Ultrasonic Laboratory, Inc. WA500E wideband
amplifier and PA620L narrow band preamplifier for our work in solid
NH_. The rf gain of the WA600E unit is 6065 db, flat to within
+3 db from 2 to 65 MHz. The detector, postamplifier circuit has a
3 MHz bandwidth and contributes an additional 20 to 25 db gain.
We now consider the coupling network and preamplifier. The prob
lems involved in coupling the sample coil to power amplifier and
receiver are manifold. There are the follov/ing three relatively in
dependent states of the system that one must contend with: 1.) The
application of an rf pulse, 2.) the time t , following the applica
tion of a pulse, during which the electronic system recovers from the
pulse and 3.) the period of observation of the FID.
)
146
In the ideal case one would have the rf pulse V (t) conform to an
equation of the form,
sin(a) t + 4>) It I < —
° (5.11)
V (t) =
P
I Nit
o
where w is the Larmor frequency corresponding to the center of the
absorption line and N is the ntimber of complete cycles in the pulse.
Let (j)=0 for sake of simplicity. The ubiquitous Heisenberg uncertainty
principle, when applied to this situation, predicts that the uncertainty
in frequency w of our pulse is roughly the inverse of the length of the
pulse. Cast in perhaps more concrete terminology, that of a frequency
spectrum of V (t) , the uncertainty in frequency becomes more apparent.
We appeal to the technique of Fourier transform analysis to transform
V (t) to the frequency domain, and find (87)
sin(aj  u) (Ntt/o) )
o
for large o) and cj close to to . The constant a' is a product of the
^ o o
Fourier transform and not important. The function g(a)) also has a
width of roughly the inverse of the length, in time, of the pulse.
Thus, we see that in order to excite all nuclei in a solid, with a
characteristic NMR distribution function f(a)) of width 10 Hz, our
pulse width should be less than 10 psec; and, furthermore, the pulse
should be presented relatively undistorted across the sample coil.
This condition requires that the bandwidth of the circuit seen by the
power amplifier, the circuit containing the sample coil, be greater
than 0.1 MHz. In addition, for maximum power transfer one must try to
;
147
match load impedance to that of the povrer amplifier.
One may place the sample coil in a resonant paralled tuned circuit
as shown in Fig. 35. The power amplifier is represented by a lossless
voltage generator in series with a characteristic output impedance Z .
The sample coil is represented as a pure inductive reactance L in series
with its intrinsic resistive component R . The 3 db bandwidth of a
parallel tuned circuit at resonance is represented by
B = 0) /2ttQo, (5.13)
where the coil quality factor is defined by,
u) L
Q»=^. (5.14)
s
The conditions on the bandwidth force certain limits on the quality
factor Qo of the sample coil. For example at oj = 27r x 20 x 10 , a
pulsewidth of 1 usee requires that Qo be less than 20. One must of
course also make certain that the receiver input impedance Z is much
greater than that of the sample coil network.
In the second time period, the energy stored in the L C network
must be dissipated, primarily in R . This ringing of the tank cir
cuit decays, in a resonant parallel tuned circuit, with a time constant
t = 2Q/a) = 1/ttB, i.e. ,
r o
V (t) = V(0)EXP[TrBt]. (5.15)
r
If V(0) = 100 V and ttB = 10 , it takes the ringing voltage 16 usee to
decay to the '^^ 10 yV amplitude of the signal. Clearly it is advanta
geous during the ringing period to have as small a sample coil Qo as
possible. Damping of the sample coil ringing is sometimes accomplished
via an active device, but we simply relied on decreasing the coil Q»,
via an adjustable resistance paralleled with the sample coil, to the
148
)
^^ Crossed Diode Pair
Sample
Coil ,
' i »■
rf
Power Amp.
^c
Receiver
.Fig. 35, TYPICAL PARALLEL TUNED SAMPLE COIL CIRCUIT
To rf Power Amp.
R.
#^
js Transformer
n
Fig. 36. LEL INPUT CIRCUIT
To rf Power Amp.
6922
^Rt
Fig. 37. ARENBURG INPUT CIRCUIT
J
149
point where the FID could be observed following decay of the ringing.
V/hen the voltage across the diode pairs drops below '^0.5 V, they
switch into the high impedance state; thereby effectively decoupling
the tank circuit from the power amplifier. One then has the situation
illustrated in Fig. 35, but with the diodes acting like an open switch.
We tune C to resonate with L ,and all stray reactance, with the cir
' cuit in this state. This tuning is done by applying a very small
amplitude modulated rf signal at to through the diodes, which remain in
a high Impedance state. The capacitor C and receiver are tuned to
maximize the signal. The parallel tuned resonant circuit is particular
ly desireable for detecting the nuclear signal V (t) because the vol
n
tage developed across the capacitor C is not V (t), but Q times
n o
V (t) . Clearly then, one would hope for the largest possible Q ,
consistent with FID bandwidth requirements, during detection of the
FID. One additional factor should also be taken into consideration —
for each input tube, there exists an optimum input source impedance at
each operating frequency. The reader who is interested in optimizing
the signaltonoise ratio (S/N) should consult references (88), (89),
and (90).
Follox^ing S/N optimization techniques in redesigning the Lei in
put circuit, increased the S/N by 7 db over the circuit shown in
ref. (77). The coupling circuit used with the Lei model 5182 is
^ " shown in Fig. 36. Our design incorporates the sample coil into the
actual input circuit, thereby eliminating additional circuit elements
which compromise performance for the sake of impedance matching to
a 50 fi load. This circuit provides sensitivity enough to easily
observe the pure quadrupole FID in a sample 3 cm long by 1 cm in
;
150
diameter of Hexamethylene tetramine at room temperature. The importance
of source impedance optimization cannot be overstressed when one must
contend with very weak signals. Incidentally, a recovery time of
'■^ 30 ysec was obtained with this circuit at 3.3 MHz; additional diode
pairs did not make any noticeable improvement in the recovery time.
A similar circuit was used in conjunction with the Arenburg model
PA620L preamplifier (see Fig. 37). The relatively strong proton re
sonance did net require much attention to matching source impedance for
optimum S/N. We found the S/N at 21 MHz to be much better than that
at other frequencies. Ringing vtas quite a problem due to the relative
ly short proton T^ of 10 ysec in solid NH~, and required severe damp
ing of the sample coil. Possibly one of the active coil damping cir
cuits in the literature would have helped in this instance. Total
dead time from the end of the pulse was roughly 5 ysec.
This brief discussion does not scratch the surface of a very in
teresting and challenging electronics problem. The reader who wishes
to investigate this problem in more detail is referred to the litera
ture; in particular, references (91), (92), (93), (94), (95), (96),
(97) and (98) may be of help.
5.3.2. The Continuous Wave Spectrometer
The continuous wave spectrometer was used in the determination of
linewidth and second moment. The spectrometer used is of rather con
/ ventional design and is illustrated in the operational configuration
in Fig. 38. We used the frequency swept, magnetic modulation mode
for all absorption line measurements.
The heart of the system is a slightly modified Robinson type
oscillator (99) constructed in this laboratory by P. Canepa (100) .
151
J
Voltage
Ramp
Generator
Magnet
Sample Coil
Modulations""^
Coils
Frequency
Counter
Robinson Oscillator
i.
Audio Power Amp.
PMMt
OeTtCTetv
^ "" P " " I * ft
QfrMO RC FluTl(\'
LockIn Amplifier
Audio Oscillator
Chart Recorder
Fig. 38. BLOCK DIAGRAM OF THE CONTINUOUS WAVE APPARATUS
152
This vinit is capable of the very low level rf (<1 to 100 mV) necessary
for work with samples having long T . The frequency sweep is effected
through application of slowly varying ramp voltage to a voltage vari
able capacitor. The ramp was generated through use of a Tektronix type
operational amplifier, operated in the typical integration configura
tion. An Electronics, Missiles, and Communications, Inc. model EJB
lockin amplifier accepted the detected rf (audio) output from the
Robinson oscillator, and the processes absorption signal was recorded
on a Brown chart recorder. A HewlettPackard model 201C audio oscil
lator supplied audio to both the lockin reference channel and a Ling
Electronics model TP1002 power amplifier. The Ling model TP1002
drove a Helmholtz pair of coils wrapped directly around the magnet
pole faces. The maximum field modulation amplitude possible in the
air gap of the magnet was roughly 16 G at 25 Hz and 10 G at 100 Hz.
A HewlettPackard model 524 C/D electronic counter, equipped with an
automatic frequency marker, was used to measure rf frequency.
..i^AWrtr— ^r«crf^.«— Vt*.»*^ •■ ■*_.■■'.. liJ A_>t >•■»■■ *.r^''*'»«. *,* ■ H.* "rtWaMfc^K..,!
)
BIBLIOGRAPHY
1 • Tables of Interatomi c Distances an d Configuration in Molecules and
Ions (The Chemical Society, Burlington House, W. 1, London, England,
1958).
2. L. Pauling, Tne Nat ure o f the Chemical Bond (Cornell University
Press, New York, 1948),
3. H. B. Gray, Elec t rons and Chemical Bonding (W. A. Benjamin, Inc.,
New York, 1955), pp. 129154.
4. G. Kaye and T. Laby, Tables of Physical and Chemical Constants
(John Wiley & Sons, Inc., New York, 1966), Thirteenth Edition.
5. F. Seitz and D. Turnbull, Solid State Physics , Vol. II (Academic
Press, Inc., New York, 1956), pp. f87.
6. C. P. Slichter, Princip le s of Magnet i c Resonance (Harper & Row,
Publishers, New York, 1963).
7. A. Abragam, Princ iples of Nuclear Ma gnetism (Oxford at the Claren
don Press, Amen House, London, 1961).
8. E. R. Andrew, Nucl e ar Magnetic Resonanc e (Cambridge at the Univer
sity Press, Cambridge, England, 1955).
9. N. Bloembargen, E. M. Pur cell and R. V. Pound, Phys . Rev. 73,, 679,
(1948).
10. J. H. Van Vleck, Phys, Rev. 74, 1168 (1948).
11. T. P. Das and E. L. Hahn, Nuclear Ouadru pole Resonance Spectro
scopy (Academic Press, Inc., New York, 1958).
12. I. Olovsson and D. H. Templeton, Acta. Cryst. 12_, 832 (1959).
13. C. T. O'Konski and T. J. Flautt, J. Chem. Phys. 2_7, 815 (1957).
14. S. S. Lehrer and C. T. O'Konski, J. Chem. Phys. 43_, 1941 (1965).
15. H. S. Gutowsky and S. Fujiwara, J. Chem. Phys. ^, 1782 (1954).
16. E. L. Hahn and D. E. Max^^ell, Phys. Rev. 88, 1070 (1952).
17. H. S. Gutowsky and D. W. McCall, Phys. Rev. 82, 748 (1951).
153
154
18. R. A. Bernheim and H. BatizHernandez, J. Chem. Phys. _40, 3446
(1964).
19. T. J. Swift, S. B. Marks, and W. G. Sayre, J. Chem. Phys. 44_, 2797
(1966).
20. F. Relf, Fundamentals of Statistical and Thermal Physics (McGraw
Hill Book Company, New York, 1965).
J 21. I. I. Rabi, N. F. Ramsey, and J. Schwinger, Revs. Mod. Phys. 26 ,
167 (1954).
22. H. Goldstein, Classical Mechanics (AddisonWesley Publishing Com
pany, Inc., Reading, Massachusetts, 1959).
23. F. Bloch, Phys. Rev. 70, 460 (1946).
24. R. K. Wangsness and F. Bloch, Phys. Rev. 89, 728 (1953).
j 25. P. Mansfield, Phys. Rev. 137 , A961 (1965),
26. P. Mansfield and D. Ware, Phys. Rev. 168 , 318 (1968).
27. J. G. Powles and P. Mansfield, Phys. Letters _2, 58 (1962).
28. J. G. Powles and J. H. Strange, Proc. Phys. Soc. (London) 82_, 6
(1963).
29. R. Bracewell, The Fourier Transform and its Applications (McGraw
Hill Book Company, New York, 1965).
30. J. A. Pople, Mol. Phys. 1, 168 (1958).
31. P. A. Staats and H. W. Morgan, J. Chem. Phys. 31, 553 (1959).
32. F. A. Mauer and H. F. McMurdie, Proceedings of the Meeting of
American Crystallographic Association, 1958.
33. J. W. Reed and P. M. Harris, J. Chem. Phys. 35, 1730 (1961).
34. M. T. Weiss and M. W. P. Strandberg, Phys. Rev. _83, 567 (1951).
35. R. Overstreet and W. F. Giauque, J. Amer. Chem. Soc. 59_, 254 (1937).
36. I. N. Krupskii, V. G. Manzhely and L. A. Koloskova, Phys. Stat.
Sol. 27, 263 (1968).
37. V. G. Manzhelii and A. M. Tolkachev, Sov. Phys. Solid State _8,
827 (1966).
38. F. P. Reding and D, F. Hornig, J. Chem. Phys. 19, 594 (1951).
39. R. F. Kruh and J. I. Petz, J. Chem. Phys. 41, 890 (1964).
155
! 40. J. G. Powles, Proceedings of the XlVth Collogue Ampere (North
Holland Publishing Co., Amsterdam, 1966), pp. 5265.
41. M. Bloom, Proceedings of the XlVth Collogue Ampere (NorthHolland
Publishing Co., Amsterdam, 1966), pp. 6571.
42. M. Bloom, International Summer School on NMR, Univ. of Waterloo,
1969 (Unpublished).
J
* 43. D. W. G. Smith and J. G. Powles, Molec. Phys. 10, 451 (1966).
44. R. L. Hilt and P. S. Hubbard, Phys. Rev. 134 , 392 (1964).
45. P. S. Hubbard, Rev. Mod. Phys. 33, 249 (1961).
46. P. S. Hubbard, Phys. Rev. 109 , 1153 (1958) and 111, 1746 (1958).
47. M. F. Baud and P. S. Hubbard, Phys. Rev. 170, 384 (1968).
48. R. Kubo and K. Tomita, J. Phys. Society Japan £, 888 (1954).
49. J. S. Waugh and E. I. Fedin, Sov. Phys.— Solid State 4_, 1633 (1963).
50. D. L. Vander Hart, H. S. Gutowsky and T. C. Farrar, J. Amer. Chem.
Soc. 89, 19 (1967).
51. H. S. Gutowsky and G. E. Pake, J. Chem. Phys. _18, 162 (1950).
52. S. W. Rabideau and P. Waldstein, J. Chem. Phys. 45, 4600 (1966).
53. E. R. Andrew, and R, Bersohn, J. Chem. Phys. 18_, 159 (1950).
54. P. S. Allen, J. Chem. Phys. 48^, 3031 (1968).
55. P. S. Allen and A. Cowklng, J. Chem. Phys. 49, 789 (1968).
56. S. Glough, J. Phys. Chem. 1, 265 (1968).
57. R. G. Fades, G. P. Jones, J. P. Llewellyn and K. W, Terry, Proc.
Phys. Soc. 91, 124 (1967).
58. R. G. Eades, T. A. Jones and J. P. Llewellyn, Proc. Phys. Soc. 91,
632 (1967).
) 59. P. S. Allen and S. Clough, Phys. Rev. Letters _22, 1351 (1969).
60. W. B. Moniz and H. S. Gutowsky, J. Chem. Phys. 38, 1155 (1963).
61. J. G. Powles and M. Rhodes, Mol. Phys. 12, 399 (1967).
I 62. J. G. Powles, M. Rhodes and J. H. Strange, Mol. Phys. n_, 515 (1966),
63. P. Thaddeus, L. C. Krisher and P. Cahill, J. Chem. Phys. 41, 1542
;
156
(1964).
64. G. Hermann, J. Chem. Phys. 29, 875 (1958).
65. J. Blicharski, J. Hennel, K. Krynicki, J. Mkulski, T. Waluga and
G. Zapalski, Bulletin Ampere 9_, 452 (1960).
66. H. S. Sandhu, J. Lees and M. Bloom, Can. J. Chem. 38, 493 (1960).
67. H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954).
68. P. M. Richards, Phys. Rev. 132 , 27 (1963).
69. P. S. Hubbard, Phys. Rev. 131, 275 (1963).
70. D. W. McCall, D. C. Douglass and E. W. Anderson, Phys. Fluids 4,
1317 (1961).
71. R. A. Ogg and J. D. Ray, J. Chem. Phys. 26., 1515 (1957).
72. R. A. Ogg, Jr., Disc. Fara. Soc. 17, 215 (1954).
73. J. Kronsbein, Private Discussion.
74. 0. E. Vilches and J. C. Wheatley, Rev. Sci. Instr. 32, 1110 (1961)
75. R. L. Powell, L. P. Ca}rwood and M. D. Bunch, Temperature — Its
Measurem.ent in Science and Industry 3^, 65 (1962) .
76. B. Ostle, Statistics in Research (The Iowa State University Press,
Ames, Iowa, 1966).
77. A. S. De Reggi, Ph.D. Thesis, University of Florida, 1966 (Unpub
lished) .
78. E. L. Hahn, Phys. Rev. 8£, 580 (1950).
79. S. Meiboom and D. Gill, Rev. Sci. Instr. 2£, 688 (1958).
80. K. Halbach, Phys. Rev. 119, 1230 (1960).
81. E. R. Andrew, Phys. Rev. 91, 425 (1953).
82. P. C. P. Canepa, Master's Thesis, University of Florida, 1961
(Unpublished) .
83. D. P. Ryan, Rev. Sci. Instrum. 37_, 486 (1966).
84. L. R. Holland, Rev. Sci. Instrum. 37, 1202 (1966).
85. E. A. Faulkner and D. W. Harding, J. Sci. Instr. 43_, 97 (1966).
86. R. C. Williamson, Rev. of Sci. Instr. 40, 666 (1969).
157
87. G. Arfken, Mathematical Methods for Physicists (Academic Press,
Inc., New York, 1966).
88. W. R. Bennett, Electrical Noise (McGrawHill Book Company, Inc.,
New York, 1960).
89. J. S. Bendat, Principles and Applications of Random Noise Theory
(John Wiley & Sons, Inc., New York, 1958).
) 90. J. M. Pettit and M. M. McWhorter, Electronic Amplifier Circuits
(McGrawHill Book Company, Inc., New York, 1961).
91. I. J. Lowe and D. E. Barnaal, Rev. Sci. Instr. 34, 143 (1963).
92. G. P. Jones, D. C. Douglass and D. W. McCall, Rev. of Sci. Instr.
36, 1460 (1965).
93. J. J. Spokas, Rev. Sci. Instr. 36, 1436 (1965).
94. R. A. McKay and D. E. Woessner, J. Sci. Instr. 43, 838 (1966).
95. D. E. Barnaal and I. J. Lowe, Rev. Sci. Instr. 37_, 428 (1966).
96. K. R. Jeffrey and R. L. Armstrong, Rev. of Sci. Instr. _38, 634
(1967).
97. V, L. Pollak and R. R. Slater, Rev. of Sci. Instr. 37, 268 (1966)
98. G. H. Czerlinski, Rev. of Sci. Instr. 39, 1730 (1968).
99. F. N. H. Robinson, J. Sci. Instr. 36_, 481 (1959).
100. P. C. P. Canepa, Hybrid Robinson Oscillator (Unpublished).
BIOGRAPHICAL SKETCH
James Lynn Carolan was born August 15, 1943, at Long Beach, Cali
fornia. He attended Beaumont High School in Beaumont, Texas and grad
uated in May, 1961. In September, 1961, he enrolled in Lamar State
College of Technology in Beaumont, Texas and received the degree of
Bachelor of Science with a major in physics in May, 1965. He enrolled
in the Graduate School of the University of Florida in September, 1965,
at which time he was granted a graduate assistantship in the Department
of Physics. Laboratory teaching duties and course work occupied his
time until December, 1966, at which time he elected to continue work
directed toward the degree of Doctor of Philosophy, bypassing the
degree of Master of Science. His work from December, 1966 until the '
present consisted solely of satisfying the requirements for the degree
of Doctor of Philosophy.
James L^Tin Carolan was married to the former Joan Marion Satzer in
December, 1965, and they expect their first child in September, 1969.
He is a member of Sigma Pi Sigma and a junior member of the American
Association of Physics Teachers .
158
)
This dissertation was prepared under the direction of the chairman
of the candidate's supervisory committee and has been approved by all
the members of that corarnittee. It was submitted to the Dean of the
College of Arts and Sciences and to the Graduate Council, and was ap
proved as partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
August, 1969
^ M
Dean, College/ 6f, Arts and Sciences
y
1^
C?vH^
Chairman: T. A. Scott
A. A. Broyle's
^■M¥^{/fr/.
E. H. Hadlock
Dean, Graduate School
jl. S. Rosenshein